i 


<X/(--<^ 


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STANDARD    ALGEBRA 


BY 


WILLIAM   J.    MILNE,   Ph.D.,  LL.D. 

/( 

PRESIDENT  OF  NEW  YORK   STATE  NORMAL  COLLEGE 
ALBANY,    N.Y. 


NEW  YORK  •:.  CINCINNATI  •:•  CHICAGO 

AMERICAN    HOOK     COMPANY 


CoPYRIGHf,   1908,   BT 

WILLIAM  J.  MTLNE. 

Entered  at  Stationers'  Hall,  London. 

standard  al6bbra. 


PREFACE 

Scope.  —  This  work  has  been  written  to  meet  the  require- 
ments of  colleges  and  universities  for  general  admission  and  of 
the  course  outlined  by  the  Regents  of  the  State  of  New  York 
for  both  elementary  and  intermediate  algebra.  Every  kind  of 
< question  asked  in  recent  examinations  has  been  covered. 

Method,  —  The  author  adheres  to  the  inductive  method  of 
]>resentation,  but  uses  declarative  statements  and  observations 
instead  of  questions.  These  are  followed  by  illustrative  prob- 
lems and  explanations  wliich  bring  out  the  important  jjoints 
that  should  be  emphasized,  and  the  treatment  is  rounded  out 
l»y  abundant  practice. 

Progress  is  from  the  known  to  the  related  unknown,  and  in 

lis  way  is  combined  the  student's  knowledge  of  arithmetic 
with  the  algebraic  knowledge  to  be  acquired.  New  ideas 
of  numl)er  are  introduced  whenever  the  development  of  the 
science  demands  it. 

Exercises.  —  The  number  of  exercises  is  extremely  large, 
and  the  variety  is  great.  The  concrete  work  is  well  balanced 
with  the  abstract,  so  that  both  skill  in  algebraic  processes  and 
ability  to  solve  problems  are  i)roperly  sustained. 

Problems.  —  The  problems  are  more  distinctly  related  to  real 
life  and  business  than  those  found  in  most  algebras.  Some  of 
the  traditional  problems  have  been  retained  because  they  are 
often  given  in  examinations;  and  besides,  they  are  useful  in 
developing  a  sort  of  intellectual  power.  But  the  work  con- 
tains a  large  number  of  fresh  and  interesting  problems  drawn 
from  commercial  life,  from  physics  and  geometry,  and  from 
various  to})ics  of  modern  interest.  While  the  formulcfi  of 
physics  and  of  geometry  are  employed  to  familiarize  the  pupils 
with  solutions  for  otiier  letters  than  ar,  y,  and  «,  no  attempt 
hjis  been  made  to  present  the  subject-matter  of  physics  or 
i^eometry. 

Algebraic  Representation.  —  Throughout  the  early  pai-t  of  the 
l)<)ok  there  are  sets  of  exercises  designed  to  t«3ach  algebraic 
language.  By  them  the  student  is  required  to  translate  into 
algebraic  notation  expressions  stated  in  words,  and  also  to  state 
in  words  expressions  that  are  written  with  algebraic  symbols. 


4  PREFACE 

Numerical  Substitution.  —  A  large  amount  of  work  is  given  in 
evaluating  expressions.  This  is  important  not  only  in  impart- 
ing a  better  idea  of  algebraic  language,  but  it  is  used  through- 
out the  book  in  testing  results.  Accuracy  is  thus  secured  by 
the  numerous  checks  and  tests  that  are  suggested  and  by  the 
requirement  that  roots  of  equations  be  verified.  The  student 
in  this  way  becomes  self-reliant,  and  reference  to  answers 
becomes  unnecessary. 

Graphs.  —  An  interesting  sidelight  and  adjunct  in  the  gen- 
eral solution  of  equations  is  given  by  the  presentation  of 
graphic  solutions.  They  are  not  to  be  substituted  for  the 
ordinary  methods  of  solution;  consequently,  they  are  put 
after,  rather  than  before,  the  particular  kinds  of  equations  to 
which  they  refer.  They  will  be  found  to  interest  the  student 
in  a  phase  of  algebra  which  has  relation  to  his  more  advanced 
work  in  mathematics. 

Factoring.  —  Present-day  requirements  omit  from  highest 
common  factor  and  lowest  common  multiple  the  method  by 
successive  division.  This  makes  it  imperative  that  the  stu- 
dent shall  be  well  prepared  in  the  subject  of  factoring;  con- 
sequently, the  author  has  treated  not  only  all  the  usual  cases 
fully  and  completely  with  plenty  of  practice,  but  the  factor 
theorem  is  taught,  thus  giving  the  student  a  method  of  attack- 
ing expressions  otherwise  very  difficult  to  factor.  The  sum- 
mary of  cases  presented  at  the  close  of  the  chapter  on  factoring 
will  give  the  student  unusual  power  in  this  important  subject. 
Factoring  by  completing  the  square  receives  attention  in  the 
chapter  on  quadratics. 

The-  solution  of  equations  by  facloring  is  treated  early  in 
the  book,  and  indeed  wherever  it  is  feasible  to  adopt  that 
method. 

Reviews. — Helpful  and  frequent  reviews  constitute  a  valua- 
ble feature  of  the  Standard  Algebra.  They  call  for  a  knowl- 
edge of  principles,  processes,  definitions,  and  for  the  solution 
of  abstract  exercises  and  of  problems. 

The  main  features  of  the  book  as  specified  above  will  com- 
mend it  to  those  who  are  looking  for  a  text  that  is  thoroughly 
up  to  date  in  its  matter,  clear  and  intelligible  in  its  presenta- 
tion, thorough  in  its  method  of  treatment,  and  certain  to  give 
the  student  not  only  a  scholarly  presentation  of  the  science 
but  delight  in  its  mastery. 

WILLIAM  J.  MILNE. 


CONTENTS 


Introduction 

DKFINITIONft    AND    NOTATION 

Positive  avt>  Xkcativk  Xi- 

Addition 

Subtraction 

1 1 1  \  1 1  \\ 

Multiplication  . 

Division 

IIkview 

Factoring    . 

Ill  \ii  w   OF  Factoring 

IIi(.iiK8T  Common  Factor 

Lowest  Common  Multiple 

Fractions 

1 J I  \  1 1  \\  . 

Simple  Equations 

Simultaneous  Simple  Equations 

(Jraphic  Solutions  —  Simple  Equations 

Involution  . 

Evolution     . 

TnKouY  OF  Exponents 

Radicals 


CONTENTS 


PAGR 

Imaginary  Numbers 

.     268 

Review 

.     272 

Quadratic  Equations        .... 

.     279 

Graphic  Solutions  —  Quadratic  Equations 

.     326 

Properties  of  Quadratic  Equations    . 

.     339 

General  Review 

.     350 

Inequalities 

.     361 

Ratio  and  Proportion 

.     369 

Variation 

.     385 

Progressions 

.     394 

Interpretation  of  Results     . 

.     411 

The  Binomial  Theorem    . 

.     416 

Logarithms 

.     423 

Permutations  and  Combinations 

.     443 

Complex  Numbers 

.     454 

Index    

.     458 

STAM)ARI)   ALGEBRA 


J>»C<: 


INTRODUCTION 


1.  The  basis  of  algebra  is  found  in  arithmetic.  Both  arith- 
metic and  algebra  treat  of  number,  and  the  student  will  find  in 
alj:jebra  many  things  that  were  familiar  to  him  in  arithmetic. 
In  fact,  there  is  no  clear  line  of  demarcation  between  arith- 
metic and  algebra.  The  fundamental  principles  of  each  are 
identical,  but  in  algebra  their  application  is  broader  than  it  is 
ill  arithmetic. 

The  very  attempt  to  make  these  principles  universal  leads 
to  new  kinds  of  number,  and  while  the  signs,  symbols,  and 
definitions  that  are  given  in  arithmetic  appear  in  algebra  with 
their  arithmetical  meanings,  yet  in  some  instances  they  take  on 
additional  meanings. 

To  illustrate,  arithmetic  teaches  the  meaning  of  5  —  3  and  so 
does  algebra,  but  it  will  be  seen  that  algebra  is  more  general 
than  arithmetic  in  that  it  gives  a  meaning  also  to  3—5,  which 
in  arithmetic  is  meaningless.  In  this  connection  the  student 
will  see  how  addition  does  not  always  mean  an  increase,  nor 
subtraction  a  decrease.  Arithmetic  teaches  the  meaning  of  9- ; 
liat  is,  9^  =  9  X  9.  Later  the  student  will  learn  that  algebra 
^ives  a  meaning  to  9^ ;  that  is,  9^  =  3,  one  of  the  two  equal 
factors  of  9. 

In  short,  algebra  affords  a  more  general  discussion  of  number 
and  its  laws  than  is  found  in  arithmetic. 


8  INTRODLCTION 

ALGEBRAIC  SOLUTIONS 

2.  The  numbers  in  this  chapter  do  not  differ  in  character 
from  the  numbers  with  which  the  student  is  already  familiar. 

The  following  solutions  and  problems,  however,  serve  to 
illustrate  how  the  solution  of  an  arithmetical  problem  may 
often  be  made  easier  and  clearer  by  the  algebraic  method,  in 
which  the  numbers  sought  are  represented  by  letters,  than  by 
the  ordinary  arithmetical  method. 

Letters  that  are  used  for  numbers  are  called  literal  numbers. 

3.  Illustrative  Problem.  —  A  man  had  400  acres  of  corn  and 
oats.  If  there  were  3  times  as  many  acres  of  corn  as  of  oats, 
how  many  acres  were  there  of  each  ? 

Arithmetical  Solution 
A  certain  number  =  the  number  of  acres  of  oats. 
Then,       3  times  that  number  =  the  number  of  acres  of  corn, 
and  4  times  that  number  =  the  number  of  acres  of  both  ; 

therefore,      4  times  that  number  =  400. 

Hence,  the  number  =  100,  tbe  number  of  acres  of  oats, 

and  3  times  the  number  =  300,  the  number  of  acres  of  corn. 

Algebraic  Solution 

Let  X  =  the  number  of  acres  of  oats. 

Then,  3  x  =  the  number  of  acres  of  corn, 

and  •  4  x  =  the  number  of  acres  of  both ; 

therefore,  4  x  =  400. 

Hence,  x  =  100,  the  number  of  acres  of  oats, 

and  3  a;  =z  300,  the  number  of  acres  of  corn. 

Observe  that  in  the  algebraic  solution  x  is  used  to  stand  for  "  a  certain 
number"  or  "  that  number,"  and  thus  the  work  is  abbreviated. 

4.  An  expression  of  the  equality  of  two  numbers  or  quan- 
tities is  called  an  equation. 

5  ic  =  30  is  an  equation. 


INTROUUCTIOX  9 

5.  A  question  that  can  be  answered  only  after  a  course  of 
reasoning  is  called  a  problem. 

6.  The  process  of  finding  the  result  sought  is  called  the 
solution  of  the  problem. 

Problems 

7.  Solve,  both  arithmetically  and  algebraically,  the  follow- 
ing problems : 

1.  A  bicycle  and  suit  cost  $54:.  How  much  did  each  cost, 
if  the  bicycle  cost  twice  as  much  as  the  suit  ? 

2.  Two  boys  dug  160  clams.  If  one  dug  3  times  as  many 
as  the  other,  how  many  did  each  dig  ? 

3.  Two  boys  bought  a  boat  for  $45.  One  furnished  4  times 
as  much  money  as  tlie  otlier.     How  much  did  each  furnish  ? 

4.  Find  a  number  whose  double  equals  52. 

5.  A  certain  number  added  to  3  times  itself  equals  96. 
What  is  the  number  ? 

6.  The  water  and  steam  in  a  boiler  occupied  120  cubic  feet 
of  space,  and  the  water  occupied  twice  as  much  space  as  the 
steam.     How  many  cubic  feet  of  space  did  each  occupy  ? 

7.  A  house  and  lot  cost  $3000.  If  the  house  cost  4  times 
as  much  as  the  lot,  what  was  the  cost  of  each  ? 

8.  In  a  fire  B  lost  twice  as  much  as  A,  and  C  lost  3  times 
as  much  as  A.  If  their  combined  loss  was  $6000,  what  was 
the  loss  of  each  ? 

9.  A  boy  bought  a  bat,  a  ball,  and  a  glove  for  $2.25.  If 
the  bat  cost  twice  as  much  as  the  ball,  and  the  glove  cost  3 
times  as  much  as  the  bat,  what  was  the  cost  of  each  ? 

10.  A  farmer  raised  a  certain  number  of  bushels  of  wheat, 
4  times  as  much  corn,  and  3  times  as  much  barley  as  corn. 
If  there  were  in  all  5100  bushels  of  grain,  how  many  bushels 
of  each  kind  did  he  raise  ? 


10  JNTRODUCTIOX 

11.  The  sides  of  any  square  (Fig.  1)  are  equal  in  length. 
How  long  is  one  side  of  a  square,  if  the  perimeter  (distance 
around  it)  is  36  inches  ? 


Fig.  1  Fig.  2  Fig.  3 

12.  The  length  of  each  of  the  sides,  a  and  b,  of  the  triangle 
(Fig.  2)  is  twice  the  length  of  the  side  c.  If  the  perimeter  is 
40  inches,  what  is  the  length  of  each  side  ? 

13.  The  opposite  sides  of  any  rectangle  (Fig.  3)  are  equal. 
If  a  rectangle  is  twice  as  long  as  it  is  wide  and  its  perimeter 
is  48  inches,  how  wide  is  it  ?     How  long  ? 

14.  In  a  business  enterprise  the  joint  capital  of  A,  B,  and  C 
was  $  8400.  If  A's  capital  was  twice  B's,  and  B's  was  twice 
C's,  what  was  the  capital  of  each  ? 

15.  The  owner  of  a  piano  found  that  the  annual  cost  of 
keeping  it  in  tune  and  insuring  it  against  fire  was  $  12.50,  and 
that  the  cost  of  keeping  it  in  tune  was  9  times  the  cost  of 
insuring  it.     Find  the  cost  of  each  item. 

16.  One  year  1500  violins  were  made  in  the  United  States. 
Twice  as  many  were  made  in  New  York  as  in  Massachusetts, 
and  these  two  states  made  half  of  all  that  were  made  in  the 
United  States.  How  many  violins  were  made  in  Massa- 
chusetts ?    in  New  York  ? 

17.  Messrs.  Jones,  Hollis  &  Frye  invested  $  225,000  in  a 
line  of  steamboats.  Mr.  Hollis  invested  3  times  as  much  as 
j\Ir.  Jones,  and  Mr.  Frye  5  times  as  much  as  Mr.  Jones.  How 
much  did  each  invest  ? 

18.  A  plumber  and  two  helpers  together  earned  $  7.50  per 
day.  How  much  did  each  earn  per  day,  if  the  plumber  earned 
4  times  as  much  as  each  helper  ? 


INTRODLTTION  11 

19.  Divide  21  into  three  parts,  such  that  the  first  is  twice 
the  second,  and  the  second  is  twice  the  third. 

20.  Divide  36  into  three  parts,  such  that  the  first  is  twice 
the  second,  and  the  third  is  twice  the  sura  of  the  first  two. 

21.  Three  newsboys  sold  60  papers.  If  the  first  sold  twice 
as  many  as  the  second,  and  the  third  sold  3  times  as  many  as 
the  second,  how  many  did  each  sell  ? 

22.  Henry  earned  a  certain  number  of  dollars  per  day.  With 
.")  days'  earnings  he  purchased  a  rifle,  and  with  20  days'  earn- 
ings, a  bicycle.  If  both  together  cost  $50,  how  much  did  he 
earn  per  day  ?     How  much  did  the  rifle  cost  ?  the  bicycle  ? 

23.  A  man  sold  some  ducks  for  50  cents  each,  and  the  same 
number  of  geese  for  75  cents  each.  If  he  received  $  12.50  for 
all,  how  many  of  each  did  he  sell  ? 

24.  A  and  B  began  business  with  a  capital  of  $  7500.  If 
A  furnished  half  as  much  capital  as  B,  how  much  did  each 
furnish  ? 

SriJOESTioK.  —  Let  y=the  number  of  dollars  A  furnished. 

25.  James  bought  a  ])ony  and  a  saddle  for  $60.  If  the 
saddle  cost  J  as  much  as  the  pony,  find  the  cost  of  each. 

26.  Separate  72  into  two  parts,  one  of  which  shall  be  \  of 
the  other. 

27.  Separate  78  into  two  parts,  one  of  which  shall  be  J  of 
the  other. 

28.  A  basket-ball  team  won  !<>  games,  or  J  of  the  games  it 
l)layed.     Find  the  number  of  games  it  played. 

Solution 
Lf't  X  =  the  number  of  games  it  played. 

Then,  f  a;  =  Ifi, 

:ind  \x  =  S. 

Therefore,  x  =  24,  the  number  of  games  it  played. 


12  INTRODUCTION 

29.  The  distance  by  rail  between  two  cities  is  35  miles. 
This  is  f  of  the  distance  by  boat.     Find  the  distance  by  boat. 

30.  The  United  States  sent  to  Germany  one  year  135,000 
pairs  of  shoes.  This  was  f  of  the  number  sent  the  following 
year.  How  many  pairs  of  shoes  were  exported  to  Germany 
the  second  year  ? 

31.  If  I  of  the  number  of  persons  who  went  on  an  ex- 
cursion to  Niagara  Falls  were  teachers,  and  240  teachers  went, 
find  the  whole  number  of  i)ersons  that  went. 

32.  Find  the  number  of  feet  in  the  width  of  a  street,  if  f  of 
the  width,  or  48  feet,  lies  between  the  curbstones. 

33.  During  a  mild  February,  coke  declined  y\  in  market 
price.  The  price  at  the  end  of  the  month  was  $  2.20  per  ton. 
What  was  the  price  at  the  beginning  of  the  month  ? 

34.  On  an  elevated  belt-line  railroad,  9|  minutes,  or  ^  of 
the  time  required  to  make  a  round  trip,  was  consumed  in  stops. 
Find  the  number  of  minutes  required  to  make  a  round  trip. 

35.  If  I  of  a  number  is  added  to  the  number,  the  sum  is  12. 
What  is  the  number  ? 

36.  If  i  of  a  number  is  added  to  twice  the  number,  the  sum 
is  35.     What  is  the  number  ? 

37.  The  difference  between  |  of  a  certain  number  and  |  of 
it  is  16.     What  is  the  number  ? 

38.  The  number  150  can  be  divided  into  two  parts,  one  of 
which  is  f  of  the  other.     What  are  the  pai'ts  ? 

39.  I  owe  in  all  $93  to  A,  B,  and  C.  If  I  owe  A  |  as 
much  as  C,  and  B  |  as  much  as  C,  how  much  do  I  owe  each  ? 

40.  For  every  car  load  of  iron  ore  dumped  into  a  furnace,  | 
of  a  car  load  of  coke  was  used  for  fuel  and  f  of  a  car  load  of 
limestone  was  used  for  a  flux.  In  all  450  car  loads  of  ore, 
coke,  and  limestone  were  used  per  day  in  the  furnace.  How 
much  of  each  was  used  per  day  ? 


DEFINITIONS   AND   NOTATION 


8.  A  unit  or  an  aggregate  of  units  is  called  a  whole  number, 
M  an  integer;  one  of  the  equal  parts  of  a  unit  or  an  aggregate 
t  equal  parts  of  a  unit  is  called  a  fractional  number. 

Such  numbers  are  called  arithmetical,  or  absolute,  numbers. 

9.  Arithmetical  numbers  have  fixed  and  known  values,  and 
.ire  represented  by  symbols  called  numerals;  as  the  Arabic 
Jvjures,  1,  2,  3,  etc.,  and  the  Roman  letters,  \,  V,  X,  etc. 

10.  You  have  seen  that  it  is  convenient,  in  solving  problems, 
to  use  letters  for  the  numbers  whose  values  are  sought.  So 
also,  in  stating  rules,  letters  are  used  to  represent  not  only 
the  numbers  whose  values  are  to  be  found,  but  also  the  num- 
bers that  must  be  given  whenever  the  rule  is  applied. 

For  example,  the  volume  of  any  rectangular  prism  is  equal  to 
tlie  area  of  the  base  multiplied  by  the  height.  By  using  Ffor 
volume,  A  for  area  of  base,  and  h  for  height,  this  rule  is  stated 
in  symbols,  thus : 

When     ^  =  60  and  /i  =  6,  r=60x6    =300; 

when  ^  =  36  and  /i  =  10,  F  =  36  x  10  =  360  ;  etxj. 

An  equation  that  states  a  rule  in  brief  form  is  called  a 
formula. 

In  each  particular  problem  to  which  the  above  formula 
applies,  A  and  h  represent  jvxed^  knoivn  values,  but  in  conse- 
nience  of  being  used  for  all  problems  of  this  class,  A  and  h 
represent  numbers  to  whieh  (unf  arithmetical  values  whatever 
may  be  assigned. 

13 


14  definitiojs.s  and  notation 

11.  A  literal  number  to  which  any  value  may  be  assigned  at 
pleasure  is  called  a  general  number. 

12.  A  general  number  or  a  number  whose  value  is  known  is 
called  a  known  number. 

The  general  numbers,  A  and  h,  in  the  formula  V=  Ax  h  {^  10),  are 
known  numbers  ;  so  also  are  the  numerals  Band  5. 

13.  A  number  whose  value  is  to  be  found  is  called  an  unknown 
number. 

In  3x  =  21,  x  is  an  unknown  number;  in  the  formula  for  volume, 
V=  A  X  h,  F  is  an  unknown  number ;  but  when  this  formula  is  changed 
to  the  formula  for  height,  h  =  V^  A,  V  and  A  are  known  numbers  and  h 
is  an  unknown  number, 

ALGEBRAIC  SIGNS 

14.  The  sign  of  addition  is  +,  read  'plus.^ 

It  indicates  that  the  number  following'  it  is  to  be  added  to 
the  number  preceding  it. 

a  +  b,  read  '  aphis  b,'  means  that  b  is  to  be  added  to  a. 

15.  The  sign  of  subtraction  is  — ,  read  'mimis.^ 

It  indicates  that  the  number  following  it  is  to  be  subtracted 
from  the  number  preceding  it. 

a  —  b,  read  '  a  minus  &,'  means  that  b  is  to  be  subtracted  from  a. 

16.  The  sign  of  multiplication  is  x  or  the  dot  (•),  read  'mid- 
tij^lied  by.^ 

It  indicates  that  the  number  preceding  it  is  to  be  multiplied 
by  the  number  following  it. 

a  X  6,  or  (z .  6,  means  that  a  is  to  be  multiplied  by  b. 

The  sign  of  multiplication  is  usually  omitted  in  algebra, 
except  between  figures. 

Instead  of  a  x  6,  or  «  •  6,  usually  ab  is  used.  But  3x5  cannot  be  writ- 
ten 35,  because  35  means  .SO  +  6. 

17.  The  sign  of  division  is  -r-,  read  '  divided  by.'' 

It  indicates  that  the  number  preceding  it  is  to  be  divided  by 
the  number  following  it. 

a  ^  b  means  that  a  is  to  be  divided  bv  &. 


DEFINJTIOXS   AND    NOTATION  15 

Division  may  be  indicated   also   by   writing  the  dividend 
al)ove  the  divisor  with  a  line  between  them. 
Such  indicated  divisions  are  called  fractions. 
",  sometimes  read  '  a  over  ?>,'  means  that  a  is  to  be  divided  by  b. 

18.  The  sign  of  equality  is  =,  read  •  /s  r(jn<(l  to^  or  ^ equaln.^ 

19.  Order  of  operations.  —  It  has  become  a  matter  of  agree- 
ment, or  a  custom,  among  mathematicians  to  employ  signs  of 
operation,  when  written  in  a  sequence,  as  follows  : 

When  only  -\-  and  —  occur  in  a  sequence  or  only  x  and  -^jtJie 
Itprations  are  performed  in  order  from  left  to  right. 

Thus,    -         3+4-2  +  3=    7-2+3  =  6  +  3=    8; 
also,  3  X  4  -f-  2  X  3  =  12  -  2  X  3  =  6  X  3  =  18. 

a  ^-  J!)  _  c  +  tZ  means  that  6  is  to  be  added  to  a,  then  from  this  result  c 
is  to  be  subtracted,  and  to  the  result  just  obtained  d  is  to  be  added. 

Wlien  X ,  -5-,  or  hoth^  occur  in  connection  with  +,  —^or  bothy  the 
indicated  multiplications  and  divisions  are  performed  first  unless 
otherwise  indicated. 

Thus,     7  + 10 -6- 3x4  =  7  + 10 -2x4  =  7  + 10 -8=  9. 

There  are  apparent  exceptions  to  the  established  order. 

For  example,  in  w»  -;-  a6  the  multiplication  is  considered  as  aheady 
i-t-rformed  ;  consequently,  m  -^  ab  means  that  m  is  to  be  divided  by  ah, 
not  that  m  is  to  be  divided  by  a  and  the  result  multiplied  by  6  ;  but 
m  ^  n  X  b  (the  multiplication  being  indicated  by  x  )  means  that  m  is  to 
bf  divided  by  a  and  the  result  multiplied  by  b. 

20.  The  signs  of  aggregation  are :  the  parentheses^  (  ) ;  the 
rniriihini.       ;  the  6rac^'^^s,  [  J  ;  the  braces,  \  \ -,  a.nd  the  vertical 

hnr,     . 

They  are  used  to  group  numbers,  each  group  being  regarded 
as  a  single  number. 

Thus,  each  of  the  forms  (a  +  b)c,  a-\-  b-  r,  [a  +  6]c,  {a  +  6K  and  ale 
siijnifies  that  the  sum  of  a  and  6  is  to  be  multiplied  by  c.  +  b\ 

All  o))erations  within  groups  should  be  performed  first. 

When  numbers  are  included  by  any  of  the  signs  of  aggregation,  they 
:ire  commonly  said  to  be  in  paretUhetntf,  in  a  parenthesis^  or  in  parentheses. 


16  DEFINITIONS   AND   NOTATION 

21.  The  sign  of  continuation  is  •  •  •,  read  '  and  so  on,'  or  'and 
so  on  to.' 

2,  4,  6,  8,  .  •  •,  50  is  read  '2,  4,  6,  8,  and  so  on  to  SO/ 

22.  The  sign  of  deduction  is  .-.,  read  '  therefore '  or  '  hence.' 


EXERCISES 

23. 

Read  and  tell  the  meaning  of : 

1. 

m  +  n. 

6.   2  .  3  -  4  w. 

11. 

a-\-  m-\-n. 

2. 

x  —  y. 

7.    3j9+5g. 

12. 

a  4-  (m  —  71). 

3. 

a^b. 

8.    7(1? +  2;). 

13. 

a  —  m  —  n. 

4. 

6      s 

9.     ^-^ 

b-\-  s 

14. 

a      m      n 

5.    ab  —  rs.  10.   x-{-y-^S.  15.    (a  +  m)(6  — n). 

Indicate  results : 

16.  Add  2  times  c  to  5  times  d. 

17.  Subtract  2  times  4  from  m  times  w. 

18.  Multiply  the  sum  of  x  and  y  by  2;. 

19.  Divide  v  —  w  by  r  times  s. 

20.  Find  the  product  of  2  ic  +  7  and  ?>y  —  2. 

21.  Express  the  product  of  a  and  a  -f-  6  divided  by  the  prod- 
uct of  b  and  a—b. 

22.  A  boy  had  a  apples  and  his  brother  gave  him  b  more. 
How  many  apples  had  he  then  ? 

23.  Edith  is  14  years  old.     How  old  was  she  4  years  ago? 
a  years  ago  ?     How  old  will  she  be  in  3  years  ?  in  &  years  ? 

24.  At  X  cents  each,  how  much  will  5  oranges  cost  ? 

25.  If  z  caps  cost  10  dollars,  how  much  will  1  cap  cost  ? 

26.  At  y  cents  each,  how  many  pencils  can  be  bought  for  x 
cents  ? 

27.  A  boy  who  has  p  marbles  loses  q  marbles,  and  afterward 
buys  r  marbles.     How  many  marbles  does  he  then  have  ? 


DEFINITIONS   AND  NOTATION  17 

28.  What  two  whole  numbers  are  nearest  to  9  ?  to  «,  if  a;  is 
whole  number  ?  to  a,  if  a  is  a  whole  number  ? 

29.  If  y  is  an  even  number,  what  are  the  two  nearest  even 
numbers? 

30.  A  woman  exchanged  x  dozen  eggs  for  8  pounds  of  sugar 
at  a  cents  a  pound  and  5  pounds  of  coffee  at  b  cents  a  pound. 
How  much  were  the  eggs  worth  a  dozen? 

FACTORS,   POWERS,   AND  ROOTS 

24.  Each  of  two  or  more  numbers  whose  product  is  a  given 
number  is  called  a  factor  of  the  given  number. 

Since  12  =  2  x  6,  or  4  x  3,  each  of  these  numbers  is  a  factor  of  12. 
Since  3  a6  =  3  x  a  x  6,  or  3  a  x  6,  or  3  x  a6,  or  3  6  x  a,  each  of  these 
numbers,  3,  a,  6,  3a,  3  6,  and  ah,  is  a  factor  of  3  ab. 

25.  When  one  of  the  two  factors  into  which  a  number  can 
Ix'  resolved  is  a  known  number,  it  is  usually  written  first  and 
<  ulled  the  coefficient  of  the  other  factor. 

In  6  ary,  5  is  the  coefficient  of  xy  ;  in  ax,  if  a  is  a  known  number,  it  is 
the  coefficient  of  x. 

In  a  broader  sense,  either  one  of  the  two  factors  into  which  a 
number  can  be  resolved  may  be  considered  the  coefficient^  or 
co-factor^  of  the  other. 

In  6  ox,  ax  may  be  considered  the  coefficient  of  o,  6  a  of  a;,  x  of  5  a,  etc. 

Coefficients  are  numerical,  literal,  or  mixed,  according  as  they 
are  composed  of  figures,  letters,  or  both  figures  and  letters. 

When  no  numerical  coefticient  is  expressed,  the  coefficient 
^  considered  to  be  1. 

26.  When  a  number  is  used  a  certain  number  of  times  as  a 
tiictor,  the  product  is  called  a  power  of  the  number. 

Wlien  a  is  used  ticice  as  a  factor,  the  product  is  the  second 

power  of  rt,  or  the  square  of  a ;  when  a  is  used  three  times  as  a 

ictor,  the  product  is  the  third  power  of  a,  or  the  cube  of  a; 

four  times,  the  fourth  power  of  a;  ?i  times,  that  is,  any  number 

of  times,  the  nth  power  of  a. 

milne's  stand,  alg.  —  2 


18  DEFINITIONS  AND  NOTATION 

27.  The  product  indicated  by  axaxaxaxa  may  be 
indicated  more  briefly  by  a^.  Likewise,  if  a  is  to  be  used  n 
times  as  a  factor,  the  product  may  be  indicated  by  a". 

A  figure  or  a  letter  placed  a  little  above  and  to  the  right  of 
a  number,  is  called  an  index,  or  an  exponent,  of  the  power  thus 
indicated.         ^n^    >}  >, 

'  .  "Tlie  exponent^  mdicates  how  many  times  the  number  is  to  be 

^;iiseiLas  a  factor. 

52  indicates  that  5  is  to  be  used  twice  as  a  factor  ;  a^  indicates  that  a  is 
to  be  used  3  times  as  a  factor. 

a^  is  read  '  a  square,'  or  '  a  second  power ' ;  a^  is  read  '  a  cube,'  or  ••  a 
third  power';  a*  is  read  'a  fourth,'  'a  fourth  power,'  or  'a  exponent 
4  '  ;  a"  is  read  '  a  nth,'  '  a  nth  power,'  or  '  a  exponent  w.' 

When  no  exponent  is  written,  the  exponent  is  regarded  as  1. 
6  is  regarded  as  the  lirst  power  of  5,  and  a^  is  usually  written  a. 

The  terms  coefficient  and  exj)onent  should  be  distinguished. 
ba  means  a  +  a  +  a  +  a  +  a,  but  a^  means  axaxaxaxa. 

28.  When  the  factors  of  a  number  are  all  equal,  one  of  the 
factors  is  called  a  root  of  the  number. 

5  is  a  root  of  25  ;  a  is  a  root  of  a*  ;  4  x  is  a  root  of  64  x^. 

One  of  the  two  equal  factors  of  a  number  is  its  second,  or 
square,  root ;  one  of  the  three  equal  factors  of  a  number  is  its 
third,  or  cube,  root ;  one  of  the  foar  equal  factors,  the  fourth 
root ;  one  of  the  n  equal  factors,  the  nth  root. 

29.  The  symbol  which  denotes  that  a  root  of  a  number  is 
sought  is  V,  written  before  the  number. 

It  is  called  the  root  sign,  or  the  radical  sign. 
A  figure  or  a  letter  written  in  the  opening  of  the  radical  sign 
indicates  what  root  of  the  number  is  sought. 
It  is  called  the  index  of  the  root. 
When  no  index  is  written,  the  second,  or  square,  root  is  meant. 

V^  indicates  that  the  third,  or  cube,  root  of  8  is  sought. 

Vox  indicates  the  square  root  of  ax,  and  Va— &,  the  square  root  of  a  -  &. 


DEFINITIONS   AND  NOTATION  19 


ALGEBRAIC  EXPRESSIONS 

30.  A  miniher  represented  by  algebraic  symbols  is  called  an 
algebraic  expression. 

31.  An  algebraic  expression  whose  parts  are  not  separated  by 
-I-  or  —  is  called  a  term;    as  2ar^,  —  5xyz,  and  -^. 

In  the  expression  2x2  _  5  xyz  +  —  there  are  three  terms. 

The  expression  m^a  4-  fe)  is  a  term,  the  parts  being  m  and  (a  +  6). 

32.  Terms  that  contain  the  same  letters  with  the  same  expo- 
nents are  called  similar  terms. 

3x2  and  12x2  are  similar  terms  ;  also  3(a  +  6)*  and  12(a  +  by ;  also  ax 
;ind  6x,  when  a  and  b  are  regarded  as  the  coefficients  of  x. 

33.  Terms  that  contain  different  letters,  or  the  same  letters 
w  ith  different  exponents,  are  called  dissimilar  terms. 

i")  a  and  3  by  are  dissimilar  terms  ;  also  3  d-h  and  3  ah'-. 

34.  An  al<]jebraic  expression  of  one  term  only  is  called  a 
monomial,  or  a  simple  expression. 

xy  and  3  ab  are  monomials. 

35.  An  algebraic  expression  of  more  than  one  term  is  called 
a  polynomial,  or  a  compound  expression. 

3  /^(  +  2  h,  xy  +  yz  -{-  z.i-.  and  a-  +  b'^  —  c^  4-  2  ab  are  polynomials. 

36.  A  polynomial  of  two  terms  is  called  a  binomial. 
3  a  +  2  6  and  x^  -  y^  are  binomials. 

37.  A  polynomial  of  three  terms  is  called  a  trinomial. 

a  +  b  +  c  and  Sx  —  2  y  —  z  are  trinomials. 

38.  An  expression,  any  term  of  which  is  a  fraction,  is  called 
a  fractional  expression. 

— ^  —  3  X  +  ~  is  a  fractional  expression. 


20  DEFINITIONS   AND   NOTATION 

39.  An  expression  that  contains  no  fraction  is  called  an 
integral  expression. 

5  a^  _  2  a  and  6  x  are  integral  expressions. 

Expressions  like  x^  +  |  x^  4.  i  a;  _|_  i  are  sometimes  regarded  as  integral, 
since  tiie  literal  numbers  are  not  in  fractional  form. 

Numerical  Substitution 

40.  When  a  particular  number  takes  the  place  of  a  letter  or 
general  number,  the  process  is  called  substitution. 

EXERCISES 

41.  1.    When  a  =  2  and  6  =  3,  find  the  numerical  value  of 
3  a6  ;  of  a*. 

Solutions.      3  a6  =  3  •  2  •  3  =  18  ;    also,  a-^  =  2  .  2  .  2  •  2  =  16. 

When  a  =  ^,  b  =  3,  c  =  10,  m  =  4:,  find  the  value  of : 

2.  10  a.  6.    om\  10.    a7n\  14.   i  a6l 

3.  2ab.  7.    2a'b.  11.    (abf.  15.    ibm. 

4.  3  cm.  8.    3b7n\  12.    a^b\  16.   I  abc. 

5.  6  be.  9.   Actb.  13.    a^c.  17.    Sb'^cmK 

18.  When  x  =  6,  y  =  3,  z  =  2,  m  =  0,  ti  =  4,  find  the  value  of 
■\/xyz^j    of  Srrrn. 

Solutions 

Vx^  =  V6  .  3  .  2  .  2  .  2  =  VT44  =  12. 
3  m%  =  3  .  02  .  4  =  3  .  0  •  4  =  0. 

Note.  —  It  will  be  seen  in  §  541  that  when  any  factor  of  a  product  IsO, 
the  product  is  0  ;  therefore,  any  power  of  0  is  0  ;  also  any  root  of  0  is  0. 

When  a  =  4,  6  =  2,  r  =  0,  s  =  5,  find  the  value  of : 

19.  V2a6.         22.    V7  r^s.      25.    3aVbh\        28.  6W^. 

20.  7  6V.  23.    3s*6«.         26.    ^a%s.  29.  2«6Vrl 

21.  Va?.  24.    ^/Sab.      27.    .SsVo^.       30.  V¥?b^. 


UEFINITIONS   AND  NOTATION  21 

31.    3^.  32.    '^.  33.    'ii^'.  34.    ^t?^. 

sb  aba  b^a*  6  a*6V 

35.  When  x  =  S  and  //  =  2,  find  the  value  of  ^  —  y^\   of 
(./•  — y)-;  of  x^— 2x11 -{-if . 

Solutions 
x2  -  y^  =  8 .  .3  -  2  .  2  =  {)  -  4  =  6. 

(x-y)2  =(3-2)-i  =  l.l=l. 

a;2-2a;y  +  y2  =  3.3-2.3.2  +  2.2=9-12  +  4  =  l, 

"When  a  =  5,  6  =  3,  ?/i  =  4,  «  =  1,  find  the  value  of : 

36.  a^-f-^^  39.    {n  —  iy.  42.    m— '. 

37.  (a +  6)*.  40.    11' -1.  43.    (6m)-». 

38.    !-— —  .  41.    ?7i  H -— .  44.     —  • 

m  —  2  n  m  —  2n  a—  b 

"45.  ab  —  bn  +  mb'^  -^  3  mw^ 

46.  {ab-bn^\-mb^-^^mn^. 

47.  2''m'^n-  —  abmn  h-  4  6/i  —  mW. 

48.  ^m  +  3a26-|6W-8a. 

49.  f  a%  +  J  mhi  —  |  rt6''  —  ?i*. 

50.  ambn^  —  J  6^m  H-  ^  mV^  —  ^  m*. 

51.  Show    that    2  x-{-3x  =  5x    when   a;  =  2;    when    a;  =  3. 
'     Giving  X  any  value  you  choose,  find  whether  2x-{-3x  =  5x. 

52.  Show  that  m(a  -h  6)  =  ma  +  m6  when  m  =  5,  a  =  4,  and 
I      6  =  3.     Find  whether  the  same  relation  holds  true  for  other 

values  of  ?»,  a,  and  b. 

i         53.   Show  that  (a  +  6)*  =  a^  +  2  a6  +  b^  when  a  =  3  and  6=  2. 
I      Find  whether  this  is  true  for  other  values  of  a  and  b. 

[         54.    When  a  =  0,  6  =  8,  c  =  5,  cZ  =  3,  find  the  value  of  c^rf 

f      4-  a6c  -  bd'  —  f  (wZ*. 

I  Solution 

r2rf  +  a6c  -  feff-^  -  J  ad*  =  5  .  5  .  3  -H  0  .  8  .  5  -  8  •  3  .  3  -  J .  0  .  3  .  3  . 3  .  3 
^  =  76  +  0  -  72  -  0  =  3. 


22  DEFINITIONS   AND   NOTATION 

When  x  =  6,  y  =  3,  z  =  0,  r=  2,  s  =  10,  find  the  value  of : 

55.  rx  +  yz  +  rs—  xz.  58.    ^xi^  —  ^yh  -\- 1  s-. 

56.  sx^  —  T'H  -\-  xyz  —  xy^.  59.    4''.s'?/^  -^  |.r^r^  —  \\  ^yh. 

57.  12z^-\-  r'lf  +  5  0^?/  -^  3  .s-.  60.    5  xy^  —  yVr^sr  +  |^  xsz. 

61.  «2  +  .7^  -\-z^  —  2xy-\-2 xz  -  2yz. 

62.  (x-yy-i-2(x-y){r-\-s)+(r-\-sy. 

63.  5a!  +  3r8-!-2a;x  72/  +  14af  +  V^. 

42.    Solution  of  problems  in  physics  by  substitution  in  formulae. 

(a)  If  an  automobile  goes  30  feet  per  second,  in  25  seconds 
it  will  go  25  times  30  feet. 

In  general,  the  space  (s)  passed  over  by  anything  moving 
with  uniform  velocity  (v)  in  any  given  time  (t)  is  equal  to 
the  product  of  the  velocity  and  the  time. 

This  physical  law  is  expressed  briefly  by  the  algebraic  formula, 

5  =  1^/. 

By  substitution  in  this  formula  find  the  space  passed  over : 

1.  By  a  train  in  30  sec,  uniform  velocity  48  ft.  per  sec. 

2.  By  a  launch  in  11  hr.,  uniform  velocity  8.2  mi.  per  hr. 

3.  By  a  torpedo  in  75  sec,  uniform  velocity  44  ft.  per  sec 

(6)  The  formula  for  the  space  (s)  through  which  a  freely 
falling  body  acted  upon  by  gravity  (g)  will  fall  in  t  seconds, 
starting  from  rest,  is 

Use  32. 16  or  9. 8  for  g  according  as  s  is  to  be  obtained  in  feet  or  in  meters. 

4.  How  many  feet  will  a  body  fall  from  rest  in  5  seconds  ? 

5.  A  stone  dropped  from  the  top  of  an  overhanging  cliff 
reached  the  bottom  in  4  seconds.  How  many  feet  high  was 
the  cliff  ?     How  many  meters  high  ? 

6.  A  ball  thrown  vertically  into  the  air  returned  in  6  seconds. 
How  many  meters  high  was  it  thrown  ?   (Time  of  fall  =  3  sec.) 


POSITIVE  AND  NEGATIVE  NUMBERS 


43.   For  convenience,  arithmetical  numbers  may  be  arranged 
ill  an  ascending  scale: 

0,       1,      2,      3,      4,      5,       ... 

! ! I I I I 


The  operations  of  addition  and  subtraction  are  thus  reduced 
to  counting  along  a  scale  of  numbers.  2  is  added  to  3  by  be- 
ginning at  3  in  the  scale  and  counting  2  units  in  the  ascending, 
or  additive,  direction ;  and  consequently,  2  is  subtracted  from 
3  by  beginning  at  3  and  counting  2  units  in  the  descending,  or 
siihtractive,  direction.  In  the  same  way  3  is  subtracted  from 
3.  But  if  we  attempt  to  subtract  4  from  3,  we  discover  that 
the  operation  of  subtraction  is  restricted  in  arithmetic,  inas- 
much as  a  greater  number  cannot  be  subtracted  from  a  less. 
It  this  restriction  held  in  algebra,  it  would  be  impossible  to 
subtract  one  literal  ninnber  from  another  without  taking  into 

<  ount  their  arithmetical  values.  Therefore,  this  restriction 
jiiiist  be  removed  in  order  to  proceed  with  the  general  dis- 
cussion of  numbers. 

To  subtract  4  from  3  we  begin  at  3  and  count  4  units  in  the 
utiscending  direction,  arriving  at  1  on  the  opposite,  or  subtrac- 
tive,  side  of  0.  It  now  becomes  necessary  to  extend  the  scale 
1  unit  in  the  subtractive  direction  from  0. 

To  subtract  5  from  3  we  begin  at  3  and  count  5  units  in  the 
•  h'scending  direction,  arriving  at  2  on  the  opposite,  or  sub- 
tractive,  side  of  0.  The  scale  is  again  extended;  and  in  a 
similar  way  the  scale  may  be  extended  indefinitely  in  the  sub- 
1  ractive  direction. 

23 


24  POSITIVE   AND  NEGATIVE   NUMBERS 

Numbers  on  opposite  sides  of  0  may  be  distinguished  by- 
means  of  the  small  signs  +  and  ~,  called  signs  of  quality,  or 
direction  signs,  +  being  prefixed  to  those  numbers  which  stand 
in  the  additive  direction  from  0  and  ~  to  those  which  stand 
in  the  subtractive  direction  from  0. 

The  former  are  called  positive  numbers,  tlie  latter  negative 
numbers. 

Zero  is  defined  as  the  result  of  subtracting  any  number  from 
itself.     Zero  is  neither  positive  nor  negative. 

Including  zero,  the  scale  of  algebraic  numbers  may  be 
written : 

...,     -5,     -4,     -3,     -2,     -1,     0,     +1,     +2,     +3,     +4,     +5,     ... 

\ \ \ \ \ \ I         I  I         I         I 

44.  By  repeating  +1  as  a  unit  any  positive  integer  may  be 
obtained,  and  by  repeating  ~1  as  a  unit  any  negative  integer 
may  be  obtained.  Hence,  positive  integers  are  measured  by  the 
positive  unit,  +1,  and  negative  integers  by  the  negative  unit,    1. 

Fractions  are  measured  by  positive  or  negRiive  fractional  units.  Thus, 
the  imit  of  +(|)  is  +(^)  ;  the  unit  of  -(|)  is  ~(-|). 

45.  If  +1  and  ~1,  or  +2  and  ~2,  or  any  two  numbers  nu- 
merically equal  but  opposite  in  quality  are  taken  together,  they 
cancel  each  other.  For  counting  any  number  of  units  from  0 
in  either  direction  and  then  counting  an  equal  number  of  units 
from  the  result  in  the  opposite  direction,  we  arrive  at  0. 

Hence,  if  a  positive  and  a  yiegative  number  are  united  into 
one  number,  any  number  of  the  units  or  parts  of  units  of  which 
one  is  composed  cancels  an  equal  number  of  units  or  parts  of 
units  of  the  other. 

46.  Quantities  opposed  to  each  other  in  such  a  way  that,  if 
united,  any  number  of  units  of  one  cancels  an  equal  number 
of  the  other  may  be  distinguished  as  positive  and  negative. 

Thus,  if  money  gained  is  positive^  money  lost  is  negative  ;  if  a  rise  in 
temperature  is  positive,  a  fall  in  temperature  is  negative  ;  if  west  longi- 
tude is  positive,  east  longitude  is  negative  ;  etc. 


POSITIVE  AND  NEGATIVE   NUMBERS  25 

47.  Positive  and  negative  numbers,  whether  integers  or 
i  ractions,  are  called  algebraic  numbers. 

Arithmetical  uumbers  are  positive  numbers. 

48.  The  value  of  a  number  without  regard  to  its  sign  is 
I  uUed  its  absolute  value. 

Thus,  the  absolute  vahie  of  both  +  4  and  —  4  is  4. 

ADDITION  AND  SUBTRACTION 

49.  The  aggregate  value  of  two  or  more  algebraic  num- 
bers is  called  their  algebraic  sum  ;  the  numbers  are  called 
addends. 

The  process  of  finding  a  simple  expression  for  the  algebraic 
sum  of  two  or  more  numbers  is  called  addition. 

50.  In  addition,  two  numbers  are  given,  and  their  algebraic 
sum  is  required;  in  subtraction,  the  algebraic  sum,  called  the 
minuend,  and  one  of  the  numbers,  called  the  subtrahend,  are 

iven,  and  the  other  number,  called  the  remainder,  or  difference, 
required.     Subtraction  is,  therefore,  the  inverse  of  addition. 
The  difference  is  the  algebraic  number  that  added  to  the  sub- 
lidhend  yivetf  the  minuend. 

51.  Negative  numbers  give  the  foregoing  definitions  a  wider 
Hinge  of  meaning  than  they  had  in  arithmetic.  In  algebra 
addition  does  not  always  imply  an  increase,  nor  subtraction  a 
decrease. 

Sum  of  Two  or  More  Numbers 

52.  Add:  EXERCISES 

1.     +5        -5        +5        +5        -^5        -5         -5 
11        Zl        1^        ll        II        ^        11 

Si  ciGESTioNs. — The  sum  of  2  positive  units  and  5  positive  units  is  7 
sitive  units  ;  of  2  negative  units  and  6  negative  units,  7  negative  units  ; 
^^S  45)  of  6  negative  units  and  5  positive  units,  0 ;  of  4  negative  units 
and  5  positive  units,  1  positive  unit  ;  of  -9  and  +6,  -  4  ;  etc. 


26  POSITIVE   AND  NEGATIVE   NUMBERS 

To  add  two  algebraic  numbers : 

Rule.  —  If  they  have  like  signs,  add  the  absolute  values  and 
prefix  the  common  sign ;  if  they  have  unlike  signs,  fiyid  the  differ- 
ence of  the  absolute  values  and  prefix  the  sign  of  the  numerically 
greater. 

By  successive  applications  of  the  above  rule  any  number  of  numbers 
may  be  added. 

Add: 

2.    +8  +11  -10  +12  -16  -20 

+  2  -7  +4  +8  -4  +5 


3. 


4. 
5. 
6. 

53.   Abbreviated  notation  for  addition. 

Reference  to  the  scale  of  algebraic  numbers  shows  that 
adding  positive  units  to  any  number  is  equivalent  to  count- 
ing them  in  the  positive  direction  from  that  number,  and 
adding  negative  units  to  any  number  is  equivalent  to  counting 
them  in  the  negative  direction  from  that  number.  Hence,  in 
addition,  the  signs  +  and  —  denoting  quality  have  primarily 
the  same  meanings  as  the  signs  +  and  —  denoting  arith- 
metical addition  and  subtraction.     For  example, 

+  1  means  0  -f  1  and  "1  means  0  —  1 ; 
also  +5  means  0  +  5  and  ~o  means  0  —  5;  etc. 

Hence,  in  finding  the  sum  of  any  given  numbers,  only  one  set 
of  signs,  +  and  — ,  is  necessary,  and  they  may  be  regarded 
either  as  signs  of  quality  or  as  signs  of  operation,  though  com- 
monly it  is  preferable  to  regard  them  as  signs  of  operation. 

Thus,  +5  +  +3  +  -6  may  be  written  +  5  +  3  —  6,  oro  +  3  -  6. 


+  7           -5   '          +8 

+  6 

-9          +10 

-3          -3             -9 

+  5 

+  3           -40 

+2          -8             +1 

-5 

+  2           +25 

+  10  +  -4H--()  +  -7  +  +  9. 

7. 

+  6+-9  +  +  5  +  +  3  +  -4. 

-12-}-  +  8H-  +  2-f-6-h+2. 

8. 

0  +  -7-f  +  4  +  -3  +  -4. 

-40-|--6-f  +  8  +  +  7++6. 

9. 

-2-h+3  +  +  6  +  +  8  +  -8. 

POSITIVE   AND  NEGATIVE  NUMBERS 


27 


When  the  first  term  of  an  expression  is  positive,  the  si^i  is 
usually  omitted,  as  in  the  preceding  illustration. 

If  there  is  need  of  distinguishing  between  the  signs  of 
•  luality  +  and  —  and  the  signs  of  operation  4-  and  — ,  the  num- 
bers and  their  signs  of  quality  may  be  inclosed  in  parentheses. 

Thus,  if  a  =  5,  6  =  -  8,  and  c  =  -  2,  then  a  +  6  +  c  =5+  (—  3)  + 
(_2);a-6-c  =  5-(-3)-(-2);a6c  =  5(-3)(-2);etc. 

54.  A  term  preceded  by  +,  expressed  or  understood,  is  called 
a  positive  term,  and  a  term  preceded  by  — ,  a  negative  term. 

Thus,  in  the  polynomial  3  4-  2  ft  —  5  c  the  first  and  second  terms  are 
positive  and  the  third  term  is  negative. 


EXERCISES 

55.    Write  with  one  set  of  signs  and  find  the  sum  : 

1.  +7 +  +8.  3.    -3 +-7.  5.   -6-f-3++16. 

2.  +G-h-5.  4.    +2  4- -4.  6.    +84- -^9 +  -15. 

Find  the  sum  :  ^ 

7.  10-74-4-9-64-3  +  5-164-24-11. 

8.  21  +  8-6-5  +  8-7+4  +  12-30  +  15. 

9.  17-2-3-4-6  +  8-2+40-18  +  13. 

10.  In  a  football  game  the 
bull  was  advanced  5  yards 
t iom  the  Juniors'  25-yard  line 
toward  the  Seniors'  goal,  then 
6  yai-ds,  then  —  8  yards  (i.e.  it 
went  back  8  yards),  and  so  on, 
as  shown  in  the  diagram. 
What  was  the  position  of  the  ball  after  3  plays  ?  after  4  plays  ? 
after  5  plays  ?  after  6  plays  ? 

11.  Plot  the  following  and  find  the  last  position  of  the  ball : 
On  15-yard  line;  gained  4  yards ;  gained  5  yards;  lost  2  yards; 

gained  30  yards ;  lost  6  yards ;  lost  2  yards ;  gained  12  yards. 


25 


\ 


-8 


±1S 


25 


-»-ll 


35 


45 


28 


POSITIVE   AND   NEGATIVE   NUMBERS 


Difference  of  Two  Numbers 

EXERCISES 

56.  On  account  of  the  extension  of  the  scale  of  numbers 
below  zero  (§  43),  subtraction  is  always  possible  in  algebra. 

When  the  subtrahend  is  positive,  algebraic  subtraction  is  like 
arithmetical  subtraction,  and  consists  in  counting  backward 
along  the  scale  of  numbers. 

Subtract  the  lower  number  from  the  upper  one : 

3  4  5  6  7  8  9 


-3 

-3 

-3 

-4 

-5 

-6 

-7 

^ 

J. 

2 

3 

4 

5 

6 

2. 


Observe  that  subtracting  a  positive  number  is  equivalent  to 
adding  a  numerically  equal  negative  number. 

When  the  subtrahend  is  negative,  it  is  no  longer  possible  to 
subtract  as  in  arithmetic  by  counting  backward. 

3.    Subtract  "2  from  8. 


PROCESS 


8  +  2 


10 


Explanation.  —  If  0  were  subtracted  from  8,  the 
result  would  be  8,  the  minuend  itself. 

The  subtrahend,  however,  is  not  0,  but  is  a  number  2 
units  below  0  in  the  scale  of  numbers.  Hence,  the 
difference  is  not  8,  but  is  8  +  2,  or  the  minuend  plus  the 
subtrahend  with  its  sign  changed. 


Subtract  the  lower  number  from  the  upper  one 
4. 


5. 


4 

4 

4 

4 

5 

7 

9 

0 

^1 

~2 

^3 

^6 

-7 

j^ 

-5 

-5 

-5 

-5 

-1 

-4 

-6 

_0 

21 

^2 

^6 

^ 

~1 

^5 

Principle.  —  Subtracting  any  number  is  equivalent  to  adding 
it  with  its  sign  changed. 


POSITIVE   AND  NEGATIVE  NUMBERS 


29 


Subtract  the  lower  number  from  the  upper  one : 


10 

12 

20 

16 

40 

32 

~A 

^ 

^6 

II 

^8 

IZ 

0 

-3 

-7 

-10 

-5 

-12 

~A 

-6 

^ 

~5 

-10 

-20 

4 

4 

-4 

-9 

3 

-7 

4 

^4 

_4 

_3 

l5 

_8 

5 

-3 

-5 

10 

-10 

13 

^6 

7 

^ 

-J_ 

7 

^ 

20 

44 

28 

-10 

-10 

~5 

3 

-4 

-6 

10 

10 

12 

10. 


11.  Subtract  12  from  "1. 

12.  Subtract  "4  from  14. 

13.  Subtract  11  from  "8. 


14.  From  "6  subtract  4. 

15.  From  0  subtract  "3. 

16.  From  "3  subtract  0. 


17.    From   0   subtract    -7;    from    the    result    subtract   ~4; 
then  add  "2;  add  ~3;  add  7;  subtract  11;  and  add  ~6. 

A  weather  map  for  January  16  gave  the  following  minimum 

and  maximum  temperatures  (Fahrenheit) : 


Chicago 

DULUTII 

Hblkna 

Montreal 

New  Orleans 

Nkw  Y<)KK 

Minimum 
Maximum 

24° 
30° 

-6° 

2° 

-12° 

-40 

-12° 
18^ 

64° 

76° 

20° 

42° 

18.  The  range  of  temperature  in  Chicago  was  6°.  Find  the 
range  of  temperature  in  each  of  the  other  cities. 

19.  The  freezing  point  is  32°  F.  How  far  below  the  freez- 
ing point  did  the  temperature  fall  in  Montreal  ? 

20.  How  much  colder  was  it  in  Duluth  than  in  Chicago? 
in  Montreal  than  in  New  York?  in  Helena  than  in  New 
Orleans  ? 


ADDITION 


57.  Arithmetically,      2  +  3=3  +  2. 

Ill  general,  a-\-b  =  b-{-a.      That  is. 

Numbers  may  be  added  in  any  order. 

This  is  the  law  of  order,  or  the  commutative  law,  for  addition. 

58.  Arithmetically, 

2 +3  +  5  =  (2 +  3) +  5  =  2  + (3 +5)  =  (2 +  5) +  3. 
In  general, 

a  +  5  +  c  =  (o  +  ?>)  +  c  =  a  +  (6  +c)  =  (a  +  c)  +  b.     That  is. 
The  sum  of  three  or  more  numbers  is  the  same  in  ivhatever 
manlier  the  numbers  are  groiqjed. 

This  is  the  law  of  grouping,  or  the  associative  law,  for  addition. 

59.  To  add  monomials. 

EXERCISES 

1.    Add  4  a  and  3  a. 

PKOCESS         Explanation.  —  Just  as  3  a's  and  4  a's   are  7  a's,  so  3  a 

4a  +4a  =  7a;  that  is,  when  the  monomials  are  similar  the  sum 

3  a  iway  be   obtained  by  adding,  the  numerical  coefidcients  and 

J  ^^  annexing  to  their  sum  the  common  literal  part. 

Add: 


2. 

2x 

3.       a 

4. 

—  a 

5. 

-4c 

?>x 

oa 

4a 

-3c 

6. 

4v 

7.-2/ 

8. 

\2mb 

9. 

40.^2 

-2v 

42/ 

-2mb 

-10a;2 

-7v 

-<dy 

—  6mb 

-60.^ 

10.    Add  4  a,  I  a,  —  3  a,  and  i  a. 

PROCESS 

4a+|a  —  3a+|-a  =  4a  —  3a  +  (|a  +  ia)  =  a  +  2a  =  3a. 


ADDITION  81 

Explanation.  —  By  §§  57,  58,  the  numbers  may  be  added  in  any 
order  or  grouped  in  any  manner.  For  convenience,  then,  we  may  Jirst 
add  those  with  integral  coefficients,  then  those  with  fractional  coefficients, 
and  afterward  add  these  sums,  as  in  the  process. 

11.  Add  5x,  —  I  a;,  2  x,  and  —  \x. 

12.  Add  l^abj  —  2  a6,  5^  «6,  and  —  3  ab. 

13.  Add  4  xyz,  —  J  xyz^  |  xyz,  and  —  24  xyz. 

14.  Add  -  <?de,  2J  c'de,  |  (^de,  and  -  .5  cMp.. 
Simplify : 

15.  2y-ly-5y-y  +  10y-Qy-\-Sy. 

16.  oa  — 3a  +  8a  — 10a  — 5a  — lla  +  24a. 

17.  ^hy-^by-10hy-lA:hy->(-A^hy. 

18.  8a-7>  +  Ga'6-lla«6-2a'6  +  9a''6. 

19.  li  x-y  -  J x^f  -  H^  ^y  -f  H ^f  +  ^f' 

20.  r,  (xv/Y  -  3  (a:2/)2  -  15  (xyy  +  4  (a^)^  + 13  (xyf. 

21.  (a  —  a;)  +  5  (a  —  a;)  +  7  (a  —  a;)  —  3  (a  —  x)  —  2  (a  —  a;). 

22.  3  a:(a,-2  -  2a;  +  3)  -  a;(a:2  _  2  a; ^_  3)  _|_  2  a;(ar2- 2  a; -1-3). 

23.  2(a:-l)-13(a;-l)  +  5(aj-l)-|-10(a;-l)-|-6(a;-l). 

Since  only  similar  terms  can  be  united  into  a  single  term, 
dissimilar  terms  are  considered  to  have  been  added  when  they 
liiive  been  written  in  succession  with  their  proper  signs. 

24.  Add  6  a,  —5  b,  —  2  a,  3  6,  2  c,  and  —  a. 

Soi.iTiox.  —  Sum  =  it  a  -  2  a  -  5  b  +  ii  h  +  2  c  —  a  =  S  a  -  2  b  +  2  c. 

Add: 

25.  2xy,  4  ab,  3  xy,  and  ab. 

26.  7nn,  —Scd,  —  6  win,  and  4  cd. 

27.  a,  —  6,  2  c,  —  2  a,  3  &,  and  —4  c. 

28.  2a,  26,  2c,  2d,  -a,  -36,  -c,  and -3d. 


32  ADDITION 

60.   To  add  polynomials. 

EXERCISES 

1.  Add  3  a  —  3  6  -f  5  c,  —  3  a  -f  2  6,  and  c  —  4  6  4-  2  a. 

PROCESS  Explanation.  — For  convenience,  similar  terms 

o     _  o  I    I    fr  may  be  written  in  the  same  column  (§  57). 
^       I   9  7  The  algebraic  sum  of  the  first  column  is  2  a,  of 

~~  '  the  second  —  -5  b,  of  the  third    +6c;   and  these 

Z  a      4rO  -\-     c  numbers  written  in  succession  express  in  its  sim- 

2  a  —  ob  ■}-  i)C  plest  form  the  sum  sought. 

2.  Simplify  11  a^b-7ab'-{- 2 ac'-j-10ab-4.ac'-{- 5 a'b-4.ab' 
^5ac'-{-b^-\-9ab^-7a^b-2b^-\-2ab^-Sab-6  a'b. 

PROCESS 

lla^b-7ab^-\-2ac^-\-10ab-\-    W 
5a^b-4.ab^-4.ac'  -2W 

-  7  d^b  +  9  ab-  +  5  ac^ 
-6a-b4-2ab'  -Sab 


Sa'b  +3ac2+    2ab-    W 

Rule. — Arrange  the  terms  so  that  similar  terms  shall  stand  in 
the  same  column. 

Find  the  algebraic  sum  of  each  column,  and  write  the  results  in 
succession  with  their  proper  signs. 

3.  Add  2  a- 3  5,  2 6 -3c,  5c -4a,  10a- 5&,  and  7 6 -3c. 

4.  Add  x-\-y-{-z,  X— y+z,  y  —  z  —  X,  z  —  x  —  y,  and  x~z. 

Simplify  the  following  polynomials  : 

5.  7 x— 11  y-\-'lz  —  l  z-\-ll  x—^y  +  l y  — 11  z—^x-\-y—x—z. 

6.  aH-36  +  5c  —  6a  +  (^-f46-2c  —  26-f5a  —  d  +  a—  6. 

7.  4.x^—^xy-{-^y'^-\-10xy  —  17y'^—llx^  —  Ctxy-\-12y?  —  2xy. 

8.  2xy  —  by--\-  x^y^  —  7xy-\-  Sy^  —  4: x^y^  +5  xy  -\-4:y^-\-  a^y^. 


ADDITION  33 

9.    2  ay  —  3  ac  —  4  ay  -\-  4  av  —  (>  a//  -f  5  ac  -f  1 1  a^  —  4  ac  —  ay. 

10.  T)  am  —  3  a-//i^  +  4  —  4  a//t  4-  aHi-  —  2  -f  5  -}-  a-m^  —  6 
-f-  o  <^//>/  +  2  a*m*  —  3  am  —  3. 

11.  eVic  — 5V^+;3v^~4Vaj-|-6V^  — V»— Vy  +  3Vy 

—  2  Vj^  +  Va;  +  2  V^  —  3  Vy. 

Add: 

12.  7a-36-f5c-10d,  26-|-d-3c-4e,  5c-6a  +  2d 

—  4e,  86  — 7a  — 8c  — e,  a  — 5c4-5d  +  lle,  a  — 6  +  c  +  2d-fe, 
and  5a-46-|-2c. 

13.  bx-'dy-2z,   42/-2a;  +  62,   3a-2x-4:y,  46-2« 

—  5.V,   a  —  5  6,   5y  —  6a;,   8a;  +  2y  —  5a  —  2  6,   and   6 a;  —  y 
-2z4-4  6. 

14.  2c  —  7d-|-6y},  ll?/i  —  3c  —  5n,  7n  —  2d  —  8c,  8d-3m 
-I-  10 c,  4 rZ  —  3  71  —  8  //I,  ?M  —  6  w,  and  2  m  — 3d. 

15.  4«3-2a:2_7a.^l^aj3^3^_^53._g4a^_g3^^2 
-fJa:,  2af»-2af'  +  8a;-f  4,  and  2a:«-3ar*-2a;  +  l. 

16.  (r'  +  5  a^6  +  5  a6<4-  6\  a*6  -  2a*  +  a^b^  -2h\  a^b^-Sa^l^ 
-■ia*b-a\md2a'-\-a*b-2a'b''-{-2a:'b^-3ab*  +  b\ 

17.  5a;«  -  ar' +  7a;  -  9,  4 a?"*  -  3  a^  +  6ar^-f  12,  a^- 5a:*- a; -7, 
l-ar^-a;«,  4  a;*- 10 a;2_^ 3  3j6_^ 4  and  a;«H-ar'- 3  a;*- 4  a;- 5. 

18.  3(a  +  6)  +  6(6  +  c),  5(a -h  6)  - 10(6  +  c),  2(a  4-6)-!- (6  + c), 
//  +  c)  -  (a  -h  6),  2(6  -\-c)-  10(a  +  6),  and  3(a  +  6)  -  3(6  -+-  c). 

19.  X  -h  3(a  4- 1)  —  2/,  —  (a  4- 1)  —  2  a;  +  4 y,  and  3 a;  —  4(a  + 1) . 

20.  a»-3a26c-6a62c,  0^6  -  6^  -  c^  -  3  a6c,  a62  4- 6*c  4- 6c2, 
/^6c  4- 4  a6*c  4- c»,  6»  -  a«6  -  a6«,  a» -h  6«c -f  6c2,  and  2  a62c  -  2  6c«. 

21.  .12ar'-4a;2^a._j_2,  .4  ar^  -  4  a;  4- -4  -  a;',  3ia;-.6  +  3a;* 
+  2ar»,  and  l-^a; 4-1.2 a;2-hH«*- 

22.  aa;  —  f  aa;*  —  J  aa^y  f  aa;*  —  i  aa;^  —  J  6a;y,  |  bxy  —  }  oa;*  —  J  a6, 
I  bxy  —  ^ab  +  laxj  and  2  a6  —  f  aa;  +  f  aa^. 

milne's  stand,  alo.  —  3 


SUBTRACTION 


EXERCISES 

61.    1.   From  10  a?  subtract  15  a;. 

PROCESS 

^^  Explanation. — Since  (§  56,  Prin.)  subtracting  any 

number  is  equivalent  to  adding  it  with  its  sign  changed, 
15  X  may  be  subtracted  from  10  a:  by  changing  the  sign 
of  15  X  and  adding  —15  a;  to  10  x. 


15  a; 


—  5a; 

2.  3.  4.  5.                      6. 

From     12  o  9  am  ^x-y-  24  mn^  ll(a-\-b) 

Take       5  a  21am  ISa^y'  12  nm'  21(a-\-b) 

7.  8.  9.  10. 

From     9a-\-7b  5a  +  106  10x  +  2y  Sm  +  Sii 

Take      2a  +  36  7a+46  6a;  +  4i/  2m  +  5n 

11.  12.  13.  14. 

Froml5  7?i+    n  7x-\-2y  4  a; +  4?/  Sp-\-3q 

Take  12  771  +  2  ?i  4a;  +  4?/  7x-^2y  10p-|-2^ 

15.  From  Sp-i-Sz  subtract  10 p  4-  z. 

16.  From  15  m  +  n  subtract  5  7n  +  3  ii. 

17.  From  3  aa;  +  5  by  subtract  4  a.'c  -f-  6  by. 

18.  From  8  abc  + 19  7nx  subtract  20  abc  +  7  7nx. 

19.  From  a  +  3  6  +  c  subtract  a  +  5  -f-  3  c. 

34 


SUBTRACTION  35 

20.    From  Sx  —  Sy  subtract  5x  —  7y. 

PROCESS  Explanation.  —  Since    (§  66,   Prin.)  subtracting 

8  a;  —  3 1/  any  number  is  equivalent  to  adding  it  with  its  sign 

5  a;  —  7  M  changed,  subtracting  5  x  from  8  a:  is  equivalent  to  add- 

^         ,  ing  —  5  X  to  8  X,  and  subtracting  —  7  y  from  —  3  y  is 

— —  equivalent  to  adding  +  7  y  to  —  3  y. 

Rule.  —  Cliamje  the  sign  of  each  term  of  the  subtrahendj  and 
(hi  the  resrdt  to  the  minuend. 

After  a  little  practice  the  student  should  make  the  change  of  sign 
ii-ntally. 

21.  22.  23.  24.  26. 

From       5  a  Qxy        —  9»i?i        —  13Va;        —   3(a4-6) 

Take    -2a        -3x?/        -4mn        -    5V«        -\0{a  +  h) 


26. 

27. 

28. 

From 

Am  —  '3n-{-2p 

8a-106  +  c 

3x-^2y-z 

Take 

2  m  — on—    p 
29. 

6a-   ob-c 

5x  —  4:y—z 

30. 

31. 

From 

((  —  6  4-c 

Sa^b-5ac^-\-9a^c 

r-s-\-t 

Take 

2a-\-b-c 

3a'b'h2ac'-9a'c 

r  +  s  —  t 

32.  From  5x  —  3y-\-z  take  2x  —  y-^Sz. 

33.  From  3  a^b  -{-b^-a^  take  4  aH)  -  8  a^  +  2  6^ 

34.  From  13tt-  +  562-4c*take8a--f 96-  +  10c2. 

35.  From  lox  —  3y-\-2z  subtract  3x-\-Sy  —  9z. 

36.  From  a--ab- b'  subtract  ab-2a^-2b^. 

37.  From  m*  —  mn  4-  n-  subtiact  2 m*  —  3  m,n  -f-  2  n*. 

38.  From  5  a:*  —  2  ary  —  /  subtract  2  aj*  -|-  2  a?y  —  3  /. 

39.  From  2 ax  — by —  b xy  subtract  2by  —  2ax  —  3xy, 

40.  From  4 a6  +  c  subtract  a^  —  U^ -f- a6c  +  2ab  —  2c. 


36  SUBTRACTION 

41.  From  2  a  +  c  subtract  a  —  6 -h  c. 

42.  From  2  m-\-n  subtract  7i  —  2p. 

43.  From  x-{-y  subtratjt  3  a  —  4  -|-  2/. 

44.  From  2  a;^  +  2  xy  subtract  x^  —  xy  —  y^. 

45.  From  2  a  —  2  d  subtract  a  —  h-\-c  —  d. 

46.  From  2  h  subtract  h  —  a  —  c  —  d. 

47.  From  w"  +  y?  subtract  a^  -  3  a^x  +  3  aa^  -  aj^. 

48.  From  a*  +  1  subtract  1  —  a  +  a^  —  a'^  +  a*. 

49.  From  the  sum  of  3  a^  —  2  a6  —  6^  and  3  a6  —  2  a'  subtract 
«-  —  ab  —  h^. 

50.  From  3x  —  y-\-z  subtract  the  sum  of  x  —  4:y-\-z  and 
2x-^3y-2z. 

51.  From  a  +  6  -|-  c  subtract  the  sum  of  a  —  b  —  c,b  —  c  —  a, 
and  c  —  a  —  b. 

52.  Subtract  the  sum  of  m^n  —  2  mn^  and  2  m^w  —  m^  —  w^ 
+  2  mn^  from  m^  —  n^. 

53.  Subtract  the  sum  of  2c  —  9a  —  3b  and  3?)  —  5a  —  5c 
from  b  —  3c-\-a. 

54.  From  3  5aj  +  4  ay  subtract  the  sum  of  3ay  —  4:bx  and 
bx  4-  a?/. 

55.  From  the  sum  of  1  +  cc  and  l  —  oi^  subtract  l  —  x-\-:tf  —  j^. 

56.  From  1 0.-3-10^  + 3  a; -7  subtract  iar^- |a^  +  4  a^'- 10. 

57.  From  i  7?i^  —  i  mhi  +  J  mn^  --fjU^  subtract  tt  —  m^-\-\  mn^ 
—  i  m^n. 

58.  From  5(a  -\-  b)  —  3(x  +  ?/)  +  4(7?i  +  ?i)  subtract  4(a  +  b) 
-h  2(a;  +  2/)  +  (m  +  ^O- 

59.  From  the  sum  of  3  ar^  —  2  a.'  +  1  and  2  a;  —  5  subtract  the 
sum  of  a;  —  ar^  -h  1  and  2  x^  —  4  x  +  3. 


SUBTRACTION  37 

PARENTHESES 
62.    Removal  of  parentheses  preceded  by  -h  or  — . 

EXERCISES 

1.  Remove  parentheses  and  simplify  3  a  -f  (6  -f-  c  —  2 a). 

Solution.  — The  given  expression  indicates  that  (6  +  c  —  2  a)  is  to  be 
added  to  H  a.  This  may  be  done  by  writing  the  terms  of  (6  +  c  —  2  a) 
after  :i  a  in  succession,  each  with  its  proper  sign,  and  uniting  terms. 

.-.  8a  +  (6  +  c-2a)=3a  +  ft  +  c-2a  =  a  +  6  +  c. 

2.  Remove  parentheses  and  simplify  4  a  —  (2  a  —  2  6). 

Solution.  — The  given  expression  indicates  that  ( +  2  a  —  2  6)  is  to  be 
subtracted  from  4  a.  Proceeding  as  in  subtraction,  that  is,  changing  the 
sign  of  each  term  of  the  subtrahend  and  adding,  we  have 

4a-(2a-2  6)=4a-2a  +  26  =  2a  +  2  6. 

Principles.  —  1.  A  parenthesis  preceded  by  a  pltis  sign  may 
be  removed  from,  an  expression  without  cimnging  the  signs  of  the 
terms  in  j)arenthesis. 

2.  A  parenthesis  preceded  by  a  minus  sign  may  be  removed 
from  an  expression,  if  the  signs  of  aU  the.  terms  in  parenthesis 
are  cfianged. 

Simplify  each  of  the  following : 

3.  a-f  (6-c).  10.  a-b-(c-d). 

4.  a  — (6  — c).  11.  a  — b  —  (—c-\- a). 

5.  x—(y  —  z).  12.  a—m  —  {n  —  m). 

6.  x-(  —  y-\-z).  13.  5a  — 26 -(a -26). 

7.  m  —  n  —  (  —  a).  14.  a  — (b  — c-\-a)  —  (c  —  b). 

8.  w-(?i-2a).  15.  2xy-^3f—{x^-^xy  —  f). 

9.  5x-(2x-^y).  16.  m  +  (Sm-n)  —  (2n—m)'\-n. 


38  SUBTRACTION 

When  an  expression  contains  parentheses  within  paren- 
theses, they  may  be  removed  in  succession,  beginning  with 
either  the  outermost  or  the  innermost,  preferably  the  latter. 

17.  Simplify  6  a;  -  [3  a  -  J 4  6  +  (8  ^  -  2  «)  -  3  /j  j  +  4  x]. 

Solution 
6x-[S  a  _  {4  />  +  (8  &  ~  2  a)  -  3  h}-\-  4a;] 
Prin.  1,  =C)x-[Za-{4b  +  Sb-2a-Sh]+ix^ 

Uniting  terms,         =  6x -[Sa -{9b  -  2a}+ 4x'] 
Prin.  2,  =6a:-[3a-96  +  2«  +  4x] 

Uniting  terms,         =6a-—  [5  a  —  9&  +  4a:] 
Prin.  2,  =6x-ba  +  Qb  -4x 

Uniting  terms,         =  2x  —  5a  -\-  9b. 

Simplify  each  of  the  following : 

18.  4:a  +  b  —  \x-{-4:a-\-b  —  2y-(x-\-y)\. 

19.  ab  —  ]ab  +  ac—a-(2a  —  ac)  +  (2 a  —  2 ac) j . 

20.  a-{-[y-\5  +  ia-(6y  +  3)l-(7y-4.a-l)]. 

21.  4  m  —  [p  +  3  71  -  (m  +  n)  +  3  —  (6p  —  3  7i  —  5  m)]. 


22.  a  +  26  4-(14a-56)-S6a  +  66-(5a-4a-4  6)J. 

23.  12a-;[4-36-(6  6H-3c)]4-?>-8-(5o-26-6);. 

24.  a-\-b~\—[a  +  b  —  (c-\-x)]  —  [3a  —  (c—x-^a)  —  b]-{--i<(\. 

25.  x" -[x' -(1  ~ x)^-  ]1  +[x' -(1-  x)  +  x^y^. 


26.  4.-][5y-(3-2x-2)^-[x  +  (oy-x  +  'S)^\. 

27.  ab-\5  +  x  —  (b  +  c  —  ab  -j-x)\+[x  —  (b  —  c  —  7)]. 

28.  -\Sax-[5xy-3z^-\-z-(4:xy-\-l6z-\-7ax']-\-3z)\. 

29.  1  -  X -  \1  -  X  -[1  -  X -(1  -  x)-(x -1)^- X  +  1\. 

30.  1  — ic  — Jl— [.T-l+(a5  — 1)  — (1  — a;)— a;]-f  1— a;J. 

31.  a  -(b  -  c)-[a  -  \b  -  c  -  (b  -{-  c  -  a)-{-(a-b)i-(c-  a)l]. 


SLBTKACTION  39 

63.  Grouping  terms  by  means  of  parentheses. 

It  follows  from  §  GU  that : 

Principles.  —  1.    Any  number  of  terms  of  an  expression  may 
inclosed  in  a  parenthesis  jjreceded  by  a  plus  sign  vnthout  chatig- 
,n(j  the  signs  of  the  terms  to  be  inclosed. 

2.  Any  nnmber  of  terms  of  an  expression  may  be  inclosed  in  a 
/xirenthesis  preceded  by  a  minus  sign,  if  the  signs  of  the  terms 
'/  be  inclosed  are  changed. 

EXERCISES 

64.  1.    In  a^  -I-  2  a6  +  &",  group  the  terms  involving  b. 

Solution 

a2  +  2  a6  +  62  =  a'^  +  (2  a?>  +  d'"*). 

2.  In  a^  —  a^  —  2xy  —  y-,  group  as  a  subtrahend  the  terms 

involving  x  and  y. 

Solution 

3.  In  ax^  -{-ab  +  2a^-\-2b,  group  the  terms  involving  .r-',  and 
also  the  terms  involving  b,  as  addends. 

4.  In  a^  -h  3  arb  -\-  3  ab^  4-  b^,  group  the  first  and  fourth  terms, 
and  also  the  second  and  third  terms,  as  addends. 

In  each  of  the  following  expressions  group  the  last  three 
iiiis  as  a  subtrahend: 

6.    a*-&»-26c-c*.  7.   a'-^2ab-j-b--cr-{-2cd-(E 

6.    a^-h^^2bc-c^.  8.    a--2ab-\-b--c^-2cd-d\ 

9.    In  a-  -\-2  ab-\-b^  —  4a  —  4  b  -t  4,  group  the  first  three 
terms  as  an  addend  and  the  fourth  and  fifth  as  a  subtrahend. 

Errors  like  the  following  are  common.     Point  them  out. 

10.  x'^-jf'-f .r-l  =  (.r'-l)-(a^-f a;). 

11.  x--y--\-2yz-z'  =  x^-{y^-^2yz^z^). 


40  SUBTRACTION 

65.    Collecting  literal  coefficients. 


Add: 

CA.i:,KV/10iS2> 

1.                   (XX 

2. 

bm 

3. 

—  ex 

hx 

—  cm 

-dx 

{a^b)x 

(b  —  c)  m 

-{c  +  d)x 

4.    ax 

5. 

cy 

6. 

—  mp 

nx 

-dy 

-up 

Subtract  the  lower  expression  from  the  upper  one : 

7.   mx  8.    dy  +  az  9.    ax  —  by 

7ix  ey  —  bz  2x  —  cy 

10.    a^x-\-aby  11.    mx  —  ny  12.    ax  —  5y 

b^x  -f-  aby  nx  —  my  5x-\-ay 

13.  Collect  the  coefficients  of  x  and  of  ?/  in  ax  —  ay  —  bx  —  by. 

Solution.  — The  total  coefficient  of  x  is  (a  —  b);  the  total  coefficient 
of  y  is  {—  a  —  h),  or  —{a  +  h). 

.-.  ax  —  mj  —  bx—by  =  (a  —  b)x  —  (a  +  b)y. 

Collect  in  alphabetical  order  the  coefficients  of  x  and  of  y  in 
each  of  the  following,  giving  each  parenthesis  the  sign  of  the 
first  coefficient  to  be  inclosed  therein : 

14.  ax  —  by  —  bx  —  cy  -\-dx  —  ey.  18.  bx—ey—2  ay  +  by. 

15.  5ax-{-3ay—2dx-\-7iy—5x—y.  19.  rx—ay—sx+2ey. 

16.  cx  —  2bx-i-7ay+3ax—lx— ty.  20.  xr  -\-ax  —  y-  +  ay. 

17.  bx-\-cy  —  2ax-{-by  —  ex  —  dy.  21.  o?— ay  —  ax  — if. 

Group  the  same  powers  of  x  in  each  of  the  following : 

22.  a:ii? -\-bx^ -exA-eoi^ —  dx^—fx. 

23.  o(?-\-3x^-\-3x  —  ax^ -3ax^-\-  bx. 

24.  x^  —  abx  —  x^  —  bx'  —  ex  —  mnx^  +  dx. 

25.  ax"^  —  x^  —  ax-  +  .r-  -|-  ax  —  x  —  abx^  +  x^. 


SUBTRACTION  41 

TRANSPOSITION  IN  EQUATIONS 

66.  In  an  t^iuation,  the  number  on  the  left  of  the  sign  of 
lality  is  called  the  first  member  uf  the   equation,  and  the 

iiuiuber  on  the  right  is  called  the  second  member. 

In  the  equation  x  =  0  4-  2,  a;  is  the  first  member  and  6  +  2  is  the  second. 

67.  Observe  how  (2)  is  obtained  from  (1)  in  each  of  the 
following : 

1.  -2  +  5  =  3  (1)  3.  6=    6(1) 

Adding  2        =2  Multiplying  by      2=   2 


Sums, 

5  =  5  (2) 

Products, 

12  =  12(2) 

2. 

Subtracting 
1  Remainders, 

4  +  3  =  7  (1) 
4        =4 
■,i  =  :i  (2) 

4. 

Dividing  by 
Quotients, 

8=   8(1) 
4=   4 

2=   2(2) 

The  following  principles,  illustrated  above,  are  useful  in 
solving  equations.  They  are  so  simple  as  to  be  self-evident. 
Such  self-evident  principles  are  called  axioms. 

68.  Axioms.  —  1-    If  equals  are  added  to  equals,  the  sums  are 

nil. 

-.    If  equals  are  subtracted  from  equals,  the  remainders  are  equal. 

3.  If  equals  are  multiplied  by  eqtials,  the  jyroducts  are  equal. 

4.  If  equals  are  divided  by  equals,  the  quotients  are  equal. 

In  the  application  of  Ax.  4,  it  is  not  allowable  to  divide  by  zero  (§  647). 

EXERCISES 

69.  1.    Solve  a;— 6=4  by  adding  6  to  both  members  (Ax.  1). 

2.  Solve  the  equation  a;  -f  3  =  11  by  employing  Ax.  2. 

3.  Solve  J  a;  =  10  by  employing  Ax.  3. 

4.  Solve  7  a;=  21.     Explain  how  Ax.  4  applies. 

5.  Solve  J  a;  =  16  in  two  steps,  first  finding  the  value  of  ^  a; 
by  Ax.  4,  then  the  value  of  a;  by  Ax.  3. 


42  SUBTRACTION 

Solve,  and  give  the  axiom  applying  to  each  step : 

6.  2x  =  6.  17.    ic  4-2  =  10.  28.    fm=9. 

7.  5x  =  i).  18.    z<;— 5  =  11. 

8.  4y--=S.  19.    ?o-|-l  =  12. 

9.  3y  =  9.  20.    s  — 7  =  10. 

10.  ^z  =5.  21.  9  +  ."?  =12. 

11.  12  =2.  22.  5+y  =  W. 

12.  ^v=3,  23.  10  +.i;  =  12. 

13.  8  V  =  24.  24.  11  +  X  =  15. 

14.  9r  =  54.  25.  20  +;y  =  30. 

15.  i?-  =  1.5.  26.  7?/  — 5  =  2. 

16.  }jX  =  2.o.  27.  2^  -}-3  =  9. 

70.     1.    Adding  7  to  both  members  of  the  equation 

a;-7  =  3, 

we  obtain,  by  Ax.  1,  a;  =  3 +  7. 

—  7  has  been  removed  from  the  first  member,  but  reappears 
in  the  second  member  with  the  opposite  sign. 

2.  Subtracting  5  from  both  members  of  the  equation 

X  H-  5  =  9, 

we  obtain,  by  Ax.  2,  ic  =  9  —  5. 

When  plus  5  is  removed,  or  transposed,  from  the  first  mem- 
ber to  the  second,  its  sign  is  changed. 

3.  Explain  the  transposition  of  terms  in  each  of  the  following : 


29. 

J7l  =  8. 

30. 

|a;=10. 

31. 

fa:  =  21. 

32. 

|.=20. 

33. 

t.=15. 

34. 

5m-l  = 

=  9. 

35. 

47i-f3  = 

=  7. 

36. 

6r-7  = 

5. 

37. 

15  +  3  = 

8. 

38. 

lx  +  2  = 

:6. 

2a.'-l  =  5; 
2a;  =  5Vl. 


3i«-f2  =  ll: 
3a;  =  ll 


4a;=14-3a;; 
4.x-\-3x=U. 


71.     Prtxciple.  —  Any  term   may   he   transposed  from  one 
member  of  an  equation  to  the  other,  provided  its  sign  is  changed. 


SUBTRACTION  43 

EXERCISES 

72.    1.    Solve  the  equation  2x  +  20  =  80  -  4  a;. 

PROCESS 

2a;-f-20  =  80-4aj 
2a;  +  4a;  =  80-20 
6ir  =  60 
a;  =  10 

Explanation. — The  first  step  in  solving  an  equation  is  to  collect  the 
unknown  terras  into  one  member  (usually  the  first  member)  and  the 
known  terms  into  the  other  member. 

Adding  4 a;  to  both  members,  or  transposing  —  ix  from  the  second 
nitinber  to  the  first  and  changing  its  sign,  places  all  unknown  terms  in 
the  first  member. 

Subtracting  20  from  both  members,  or  transposing  +  20  from  the  first 
member  to  the  second  and  changing  its  sign,  places  all  known  terms  in 
the  second  member. 

Tniting  similar  terms  and  dividing  both  members  by  6,  the  coefficient 

z,  we  find  the  value  of  x  to  be  10. 

Verification.  —  The  result  should  always  be  verified  by  substituting 
it  for  the  unknown  number  in  the  (jiven  equation.  If  the  members  of  the 
given  equation  reduce  to  the  same  number,  the  result  obtained  is  correct. 

Substituting  10  for  x,  makes  the  first  menjber  20  -f  20,  or  40,  and  the 
second  member  80  —  40,  or  40.     Hence,  10  is  the  true  value  of  x. 

2.   Solve  the  equation  7  —  5  a;  =  7  —  30. 

FIRST    PROCESS  SECOND    PROCESS 

7_5a;  =  7-30  ;-6a;  =  ;r-30 

-6a;= -30  *30  =  5a; 

5a;  =  30  6  =  arora;  =  6 

a;  =  6  Test.     7-56  =  7-30 

Suggestions.  —  1.  By  Ax.  2  the  same  number  may  be  subtracted,  or 
rrinceled^  from  both  members. 

2.  By  Ax.  2  the  signs  of  all  the  terms  of  an  equation  may  be  changed 
(first  process) ;  for  each  member  may  be  subtracted  from  the  correspond- 
ing member  of  the  equation  0  =  0. 

3.  To  avoid  multiplying  by  -  1,  the  second  proceas  may  be  adopted. 


u 


[ 

SL'BTliACTJOJ 

Solve  and  verify : 

3. 

3  =  5-0^. 

12. 

8+7a  =  5a  +  20. 

4. 

9-5x=  -1 

13. 

2  +  13/1  =  50-9. 

5. 

10  +  V  =  18  -  V. 

14. 

50  -  n  =  20  +  n. 

6. 

2^4-2  =  32-2. 

15. 

3x-2S  =  x-n. 

7. 

7x-\-2  =  x-\-U. 

16. 

4i«  +  12  =  2ie-|-36. 

8. 

3p  +  2  =;>4-30. 

17. 

2x-\-^x  =  30-ix. 

9. 

om  —  5  =  2m4-4. 

18. 

Sx-\x=30-{-ix. 

10. 

6y-2  =  Sy-h7. 

19. 

5iv-10  =  ^iv-hl6. 

11. 

8a;-7  =  3  +  6x. 

20. 

4r-18=20  +  ir. 

Simplify  as  much  as  possible  before  transposing  terms,  solve, 
and  verify  : 

21.  10  ic  +  30  -4  ft.'-  (9  X  -  33  -  12  a;)  =  90  +  12  -  4  a;. 

22.  16  X  +  12  -  75  +  2  a;  - 12  -  70  =  8  re  -  (50  +  25). 

23.  11  s  -  (50  +  5  .s  +  17  -  (2  s  -  41)  -  3  s  =  2  s  +  97. 

24.  102  -  35  -  (12 2  -  16)  +  32  =  4  2  -  (35  -  10  2)  +  32. 

25.  36  +  5  X  -  22  -  (7  X  -  16)  =  5  a;  +  17  -  (2  ic  +  22). 

26.  12  a;  -  (6  if -17  X -15 -.'c)  =15- (2-17  ic  + 6  a; -4-12  a;). 

27.  14  n  -  35  =  9  -  (11  w  -  4  -  16  + 10  Ti  -  n)  + 136  - 16  n. 

Algebraic  Representation 

73.     1.    Express  the  sum  of  x,  —  y,  and  —  z. 

2.  AVhat  number  is  n  less  than  25  ? 

3.  Express  the  number  that  exceeds  a  by  6. 

4.  How  much  does  z  exceed  10  +  ?/  ? 

5.  What  number  must  be  added  to  m  so  that  the  sum  shall 
be  4? 

6.  Mary  read  10  pages  in  a  book,  stopping  at  the  top  of 
page  a.     On  what  page  did  she  begin  to  read  ? 


SUBTRACTION  45 

7.  A  man  made  three  purchases  of  a,  6,  and  2  dollars, 
K'spectively,  and  tendered  a  lO-dollar  bill.  Express  the 
number  of  dollars  in  change  due  him. 

8.  A  has  X  dollars  and  B,  y  dollai-s.  If  A  gives  H  m  dollars, 
how  much  will  each  then  have  ? 

9.  If  40  is  separated  into  two  parts,  one  of  which  is  x,  rep- 
sent  the  other  part. 

10.  What  two  whole  numbers  are  nearest  to  x,  if  a;  is  a  whole 
number  ?   to  a  -f-  b,  if  a  -f  6  is  a  whole  number  ? 

11.  If  a;  is  an  even  number,  what  are  the  two  even  numbers 
luarest  to  a;?  the  two  odd  numbers  nearest  to  aj  ? 

12.  If  n  -f  2  is  an  odd  number,  what  are  the  two  odd  num- 
bers nearest  to  7i  -f  2  ?   the  two  even  numbers  ? 

13.  There  is  a  family  of  three  children,  each  of  whom  is  2 
years  older  than  the  next  younger.  When  the  youngest  is  x 
years  old,  what  are  the  ages  of  the  others  ?  When  the  oldest 
is  //  years  old,  what  are  the  ages  of  the  others? 

14.  A  boy  who  had  x  dollars  spent  y  cents  of  his  money. 
1  low  much  money  had  he  left? 

15.  The  number  25  may  be  written  20  +  5.  Write  the  num- 
Ikt  whose  first  digit  is  x  and  second  digit  y, 

16.  The  number  376  may  be  written  300  -f-  70  +  6.  Write  the 
number  whose  first  digit  is  a;,  second  digit  y,  and  third  digit  z. 

SOLUTION   OF   PROBLEMS 

74.  If  3  a;  =  a  certain  number  and  «  + 10  =  the  same  number, 
then,  3x  =  x-\-10. 

This  illustrates  another  axiom  to  be  added  to  the  list  in  §  68. 
It  will  be  found  useful  in  the  solution  of  problems. 

Axiom  5.  —  Numbers  that  are  equal  to  the  same  iHnnhc,-.  "r  t'> 
<  '{nal  numbers^  are  equal  to  each  other. 


46  suiri'UAcriON 

75.  Illustrative  Problem.  —  Of  the  steam  vessels  built  on  the 
Great  Lakes  one  year,  21,  or  5  less  than  ^  of  all,  were  of  steel. 
How  many  steam  vessels  were  built  on  the  Lakes  that  year  ? 

Solution.  —  Let  x  —  the  number  of  steam  vessels  built. 

Then,  \x  —  h  —  the  number  of  steel  vessels. 

But  21  =  the  number  of  steel  vessels. 

.".  Ax.  5,  |x  —  5  =  21.  ^ 

Transposing,  |x  =  21  +  5  =  20. 

Hence,  Ax.  3,  x  =  78,  the  number  of  steam  vessels  built. 

76.  A  problem  gives  certain  conditions,  or  relations,  that 
exist  between  known  numbers  and  one  or  more  unknown 
numbers.  The  statement  in  algebraic  language  of  these  con- 
ditions is  called  the  equation  of  the  problem. 

The  equation  of  the  problem  just  solved  is  ix  —  6  =  21. 

77.  General  Directions  for  Solving  Problems.  —  1.  Represent 
one  of  the  imk)ioicn  namhers  by  aonie  letter,  as  x. 

2.  From  the  conditions  of  the  problem  find  an  expression  for 
each  of  the  other  unknoivn  numbers. 

3.  Find  from  the  conditions  two  expressions  that  are  equal  and 
ivrite  the  equation  of  the  problem. 

4.  Solve  the  equation. 

Problems 

78.  1.    What  number  increased  by  10  is  equal  to  19  ? 

2.  AVhat  number  diminished  by  30  is  equal  to  20  ? 

3.  What  number  diminished  by  111  is  equal  to  — 15  ? 

4.  What  number  exceeds  \  of  itself  by  10  ? 

5.  Five  times  a  number  exceeds  o  times  the  number  by  14. 
What  is  the  number  ? 

6.  If  5  is  subtracted  from  a  certain  number,  and  the  differ- 
ence is  subtracted  from  3  times  the  number,  the  result  is  35. 
What  is  the  number  ? 

7.  The  double  of  a  number  is  64  less  than  10  times  the 
number.     What  is  the  number  ? 


SUBTRACTION  47 

8.  The  sum  of  a  number  and  .04  of  itself  is  46.8.     What  is 
the  number  ? 

9.  What  number  decreased  by  .35  of  itself  equals  52  ? 

10.  If  4  is  subtracted  from  a  certain  number,  and  the  differ- 
ence is  subtracted  from  40,  the  result  is  3  times  the  number. 
What  is  the  number  ? 

11.  Three  times  a  certain  number  is  as  much  less  than  72  as 
I  times  the  number  exceeds  12.     What  is  the  number  ? 

12.  Twice  a  certain  nuinber  exceeds  ^  of  the  number  as  much 
-  (>  times  the  number  exceeds  65.     What  is  the  number  ? 

13.  A  prime  dark  sea-otter  skin  cost  $400  more  than  a 
blown  one.  If  the  first  cost  3  times  as  much  as  the  second, 
how  much  did  each  cost  ? 

14.  The  total  height  of  a  certain  brick  chimney  in  St.  Louis 
is  172  feet.  Its  height  above  ground  is  2  feet  more  than  16 
limes  its  depth  below.     How  high  is  the  part  above  ground? 

15.  A  wagon  loaded  with  coal  weighed  4200  pounds.  The 
coal  weighed  1800  pounds  more  than  the  wagon.  How  much 
did  the  wagon  weigh  ?  the  coal  ? 

16.  It  cost  a  mine  owner  $1.90  per  ton  to  mine  soft  coal 
and  ship  it  to  market.  The  cost  of  shipping  the  coal  was 
$  .10  more  per  ton  than  the  cost  of  mining  it.  Find  the  cost 
of  mining  it. 

17.  A  mining  company  sold  copper  ore  at  $5.28  per  ton. 
The  profit  per  ton  was  $  .22  less  than  the  expense.  What  was 
t  lie  profit  on  each  ton  ? 

18.  Recently  Germany  had  1025  ships  of  over  1000  tons 
<  apacity.  There  were  25  more  than  \  as  many  sailing  vessels 
as  steamers.     Find  the  number  of  each. 

19.  The  length  of  the  steamship  LusUania  is  790  feet,  or  2 
feet  less  than  9  times  its  width.  What  is  the  width,  or  beam, 
"f  the  vessel  ? 


48  SUBTRACTION 

20.  The  Canadian,  or  Horseshoe,  Falls,  in  the  Niagara  River, 
are  158  feet  high.  This  is  8  feet  more  than  i^  of  the  height 
of  the  American  Falls.  Find  the  height  of  the  American 
Falls. 

21.  The  width  of  the  St.  Lawrence  River  at  Quebec  at  a 
point  where  it  is  spanned  by  a  bridge  is  1800  feet.  This  is 
180  feet  less  than  f  of  the  length  of  the  bridge.  How  long 
is  the  bridge  ? 

22.  An  American  cargo  consisting  of  24,470  barrels  and 
boxes  of  apples  was  landed  at  Bremen,  Germany.  There  were 
170  less  than  15  times  as  many  barrels  as  boxes.  Find  the 
number  of  barrels  and  the  number  of  boxes. 

23.  In  lighting  a  hall  a  certain  number  of  16-candle  power 
electric  lamps  and  twice  as  many  20-candle  power  lamps  were 
used.  The  total  illumination  amounted  to  224  candle  power. 
Find  the  number  of  lamps  of  each  kind  used. 

24.  The  principal  sources  of  the  world's  cotton  crop  one 
year  were  the  United  States  and  India,  which  together  sup- 
plied 16,700,000  bales.  The  former  supplied  50,000  bales 
more  than  3^  times  as  much  as  the  latter.  How  much  cotton 
did  each  country  produce  that  year  ? 

25.  At  the  waterworks  2  large  pumps  and  4  small  ones  de- 
livered 4800  gallons  of  water  per  minute.  Each  of  the  lai'ge 
pumps  delivered  4  times  as  much  water  as  each  small  pump. 
How  many  gallons  per  minute  did  each  small  pump  deliver  ? 
each  large  pump  ? 

26.  The  cost  of  construction  for  4  miles  of  macadam  road 
was  ^2400  more  than  for  8  miles  of  gravel  road.  The  latter 
cost  f  as  much  per  mile  as  the  former.  What  was  the  cost  of 
each  per  mile  ? 

27.  The  total  duty  on  a  quantity  of  figs,  raisins,  and  dates, 
imported  into  the  United  States,  was  ^50.  The  duty  on  the 
raisins  was  1^  times  as  much  as  on  the  figs,  and  that  on 
the  dates  \  as  much  as  on  the  figs.  What  was  the  duty  on 
the  figs  ?  the  raisins  ?  the  dates  ? 


REVIEW  49 

REVIEW 

79.  1.  Define  square  of  a  number ;  square  root  of  a  number. 
Show  the  difference  between  these  by  an  illustrative  example. 

2.  Distinguish  between  power  and  exponent;  between 
exponent  and  coefficient. 

3.  By  what  law  do  we  know  that  the  sum  of  2  x,  —  3  ?/,  4  x, 
and  5 y  is  the  same  as  the  sum  ot2xj4x,5 y,  and  —3y ? 

4.  From  the  sum  of  z  +  xy  and  z  —  x^y  subtract  the  sum  of 
./•//  —  ZfX^  —  x^y,  and  y^  —  a?. 

5.  From  the  sum  of  '6 y?  —  2 xy  -\- y^  and  2xy  —  5y  subtract 
-  j^—6icy  +  4y^  less  x^  —  xy-^y'. 

6.  What  number  added  to  —a--{-b^  —  2ab  gives  0  for  the 
sum  ? 

7.  Give  the  general  name  that  belongs  to  the  two  following 
expressions,  and  the  specific  name  of  each : 

x^  —  Qcy  +  y^  and  x^  —  Q?y. 

8.  Remove  parentheses  and  then  regroup  the  terms  in 
order,  first  as  binomials;  second  as  trimonials  : 

2a-{2x^  +  2h-c-\-2d-{2f-z)-\-m\']-\-n\ 

9.  Inclose  the  last  four  terms  oi  a  —  y  —  h-{-x  —  2  m  brack- 
ets and  the  last  two  terms  in  parentheses. 

10.  Collect  similar  terms  within  parentheses: 

aa?  —  cy-\-ax—2  ax^  -\- 2  cif  —  ax  —  c}f -\-  ax^  -\-  cy. 

11.  How  does  kl  -»-  yz  differ  in  meaning  from  k  xl-^-y  xz? 

12.  Define  positive  numbers;  negative  numbers.    Illustrate. 

13.  The  temperature  at  White  River  Junction  one  winter 
day  was  —  40°F.  The  temperature  at  Washington  that  day 
was  40°  F.     What  was  the  difference  in  temperature  ? 

What  is  the  difference  in  the  absolute  value  of  the  numbers 
denoting  these  two  temperatures?  in  their  relative  value? 

MILNE^S   STAND.   ALO.  —  4 


50  REVIEW 

When  a  =  1, 6  =  2,  c  =  3,  c?  =  4,  and  e  =  5,  find  the  value  of : 

14.  a  —  (e-\-b)  —  (c  +  d)  —  (e  —  d-^b-{-  c). 

15.  Sab'^-2  he"  -  (cfe"  -  ac^)  +  8  he\ 

16.  5ac  +  5V^-4-a-|-2(a  — 6)(d  — e)  4-6ce. 

17.  V2  ed6  +  4e  -  -^9  a'c^  —  2  c'-d  —  (a?>e  —  abcde), 

18.  Show  that  a  number  may  be  transposed  from  one  mem- 
ber of  an  equation  to  the  other,  if  its  sign  is  changed. 

Solve,  giving  reasons  for  each  step : 

19.  5a;  +  5  =  7a^  — 3. 

20.  2  ic  -  (4  +  ir)  -  5  a;  +  20  =  4  a?  +  (4  -  5  a;). 

21.  3x-5-6ic  +  l-(9a;-5-5a;)  =  3a!-9. 

22.  10.T-3-(4-2a:)  +  (3x'-4a;  +  5-2a;)  =  2-3x  +  4a; 
-{2x  +  x)  +  l. 

23.  Prove  that  5  .t  +  4  =  6  a^  —  1,  when  a;  =  5. 

24.  Prove  that  17  —  a?  =  a.-  —  19,  when  x  =  18. 

25.  Show    that   x(x  —  y-[-  xy)  =  x-  —  xy-\-  sry,   for   as  many- 
numerical  values  as  you  may  substitute  for  x  and  y. 

Supply  the  missing  coefficients  in  the  following  equations : 

26.  ^a-'^h-\-Q>a  +  bh-*xy  =  *a  +  h-2xy. 

27.  7?-{-2xy-\-Zy^-[_2x''  +  *y'^^  =  *x^  +  i'xy. 

28.  6  m^H-9  mn— 3  w^— [3  m--\-*  mn']-\-*  n-  =  *m^—mn—2 n^. 

29.  From  a^"'b''  +  arb^""  -  a^"  take  -  2  a^'"^"  +  a"^". 

30.  If  a;  =  »'2  +  rs-s2,2/  =  2r2  +  47-s  +  2s2,  andz  =  9'2-3rs-s2, 
find  the  value  oi  x  -\-  y  —  z. 

31.  Add  .5  m^  +  2.5  m-n  -h  w  +  3,  —  .6  mw^  -f  .5  m^  +  .5  n  —  3, 
and  —  m^  —  2.5  m^n  — 1.5  w  + 1. 

32.  Add  .5x'-S^f-{-iz\  -3iz^-4.x'-{-.5f,  -if-z' 
+  a;^  and  9|  «*  -  IJ  2'  +  2.25  y\ 


MULTIPLICATION 


80.  As  in  arithmetic,  the  number  multiplied  is  called  the 
multiplicand;  the  number  by  which  the  multiplicand  is  multi- 
l>lied,  tiie  multiplier;  and  the  result,  the  product. 

When  the  multiplier  is  a  positive  integer,  multiplication 
may  be  defined  as  the  process  of  taking  the  multiplicand 
;i(lditively  as  many  times  as  there  are  ones  in  the  multiplier. 

Thus,  since  3  =  1  +  1  +  1,    5.3  =  5  +  5  +  5=15. 

Since  fractional  and  negative  multipliers  cannot  be  obtained 
l)y  simple  repetitions  of  1,  the  definition  of  multiplication  must 
now  be  stated  in  more  general  terms. 

Although  fractional  and  negative  multipliers  cannot  be  ob- 
tained directly  from  the  prhruiry  unit  1,  they  may  be  obtained 
fiom  units  derived  from  1,  by  taking  a  part  of  1,  or  by  chang- 
ing the  sign  of  1,  or  by  both  operations. 

Thus,       |  =  +  j  +  j  +  iaud-i  =  (-i)  +  (-i)  +  (-i). 

In  multiplying  5  by  3,  we  first  observe  how  3  is  derived 
from  the  primary  unit  1 ;  then  in  this  process  we  replace  1  by  5. 

Therefore,  in  multiplying  5  by  |,  in  order  to  extend  the 
definition  of  multiplication  in  harmony  with  the  existing  one, 
having  observed  that  the  multiplier  J  is  derived  from  the 
primary  unit  by  taking  3  of  the  4  equal  parts  of  it,  we  should 
take  3  of  the  4  equal  parts  of  the  multiplicand  5. 
Ihiw,  8ince  |  =  ^  +  i  +  j,    6.J  =  J+J  +  |  =  >^.     Similarly, 

:;  i)4.(_i)  +  (_i);    ...5-(-3)  =  (-5)  +  (-5)  +  (-5)  =  -15. 

M'l/h'jth'ration  is  the  procesn  of  fin.dimj  a  number  that  is  oh- 
f'lined  Jrom  th('  ni"1tip1lrn},f1  jnnt  as  the  midtiplier  is  obtained 
from  xinity. 

51 


52  MULTIPLICATION 

81.  Arithmetically,     2x3  =  3x2. 

Ill  general,  ah  =  ha.     That  is, 

The  factors  of  a  product  may  he  taken  in  any  order. 
This  is  the  law  of  order,  or  the  commutative  law,  for  miiltiijli- 
cation. 

82.  2  X  3  X  5  =  (2  X  3)  x  5  =  2  x  (3  x  o)  =  (2  x  5)  x  3. 
In  general,  ahc  =  {ah)  c  =  a  (he)  =  (ac)  h.     That  is. 

The  factors  of  a  product  may  he  grouped  in  any  manner. 
This   is   the   law   of   grouping,   or   the   associative   law,    for 
multiplication. 

83.  Sign  of  the  product. 

(1)  Suppose  that  the  multiplier  is  a  positive  number,  as  -\-  2. 
Since  +  2  may  be  obtained  from  -f  1  by  taking  -\- 1  addi- 

tively  2  times,  a  process  that  involves  no  change  of  sign,  by  the 

definition  of  multiplication  any  number  may  be  multiplied  by 

-f  2  by  taking  the  number,  icith  its  own  sign,  additively  2  times. 

+  4.2=:(  +  4)+(  +  4)=+8;  -4.2  =  (-4)  +  (-4)  =  -8. 

The  product,  therefore,  has  the  same  sign  as  the  multiplicand. 

(2)  Suppose  that  the  multiplier  is  a  negative  number,  as  —  2. 
Since  —  2  may  be  obtained  from  + 1  by  changing  the  sign  of 

+  1  and  taking  the  result  additively  2  times,  any  number 
may  be  multiplied  by  —2  by  changing  the  sign  of  the  number 
and  taking  the  result  additively  2  times. 

+  4. -2  =  (-4) +  (-4)  =-8;         _  4  .-2=  (+ 4)  +  (  +  4)  =  +8. 

The  product  has  the  sign  opposite  to  that  of  the  multiplicand. 

(3)  The  conclusions  of  (1)  and  (2)  may  be  written  as  follows  : 
From  (1),         +  a  multiplied  by  -\-h=  -\-  ah, 

and  —a  multiplied  by  -\-h=  —  ah; 

from  (2),  +  a  multiplied  by  —  5  =  —  ah, 

and  —  a  multiplied  by  —h=+  ah.         Whence, 

84.  Law  of  Signs  for  Multiplication.  —  The  sign  of  the  prroduct 
of  two  factors  is  +  when  the  factors  have  like  signs,  and  —  when 
they  have  unlike  signs. 


Mll/ni'LICATION  58 

EXERCISES 

85.    1.    Multiply  each  of  the  following  by  -|-  2 ;  then  by  —  2  : 
3,  5,   —  6,  10,   —  8,   —  9,   —  12,  a,  x,  —  h. 


2.    Multiply 

Hy 

-8 
6 

9 
8 

6 
—  5 

4 

-7 

-   2 
10 

3.    Multiply 
By 

a 
4 

-b 
6 

—  X 

-8 

-y 
-1 

n 
-12 

86.  Product  of  two  monomials. 

Tlie  product  of  two  numbers  must  contain  all  the  factors, 
numerical  and  literal,  of  both  numbers.  These  may  be  taken 
in  any  order  or  associated  in  any  manner  (§§  81,  82). 

Usually  the  numerical  coefficients  are  grouped,  to  form  the 
coefficient  of  the  product;  then  the  literal  factors  are  written, 
any  like  factors  that  may  exist  being  grouped  by  exponents. 

Thus,  §  27,  4  m^n  •  3  m^n'^  =  i  •  m  •  m  •  n  -  -i  -  m  •  in  •  m  •  n  •  n 
§  82,  =  (4  •  3)( w»  •  m  •  w  •  in  •  in)  (n  •  n  •  n) 

§  27,  =12  jn*7?\ 

87.  Law  of  Coefficients  for  Multiplication.  —  The  coefficient 
of  the  product  is  equal  to  the  jn'odnrt  (ff  the  coefficients  of  mult ipli- 
<and  and  multiplier. 

88.  Law  of  Exponents,  or  Index  Law.  for  Multiplication.  —  TVie 
"j'lKinent  of  a  uamber  in  the  jn'oduct  is  equal  to  the  sum  of  its 

I'ponents  in  multiplicand  and  multiplier. 

The  proof  for  positive  integral  exponents  follows : 

I^t  //J  and  n  be  any  positive  integers,  and  let  a  be  any  number. 
By  notation,  §  27, 

a*  =  a  •  a  •  rt  ..•  to  m  factors, 
and  a*  =  a  •  rt  •  a  •••  to  n  factors ; 

.  fjm  .  fjn  —  ^a  '  a  •  a  •••  to  m  factor8)(a  •  «  •  a  ••  to  n  factors) 

}>y  assor.  law,  =  a  ■  ti  ■  n    ■    \i<  (m  -\-  n)  factors 

1)V  notation,  =  «"••*■". 


54  MULTIPLICATION 

EXERCISES 

89.    1.   Multiply  -  4  aar^  by  2  aV. 

Explanation.  —  Since  the  signs  of  the   monomials  are 
PROCESS     unlike,  the  sign  of  the  product  is  minus  (Law  of  Signs). 

—  4  ax^  4.2  =  8  (Law  of  Coefficients). 

2  a^x'*  a  '  a^  =  a^ '  a^  =  a^+^  =  a^  (Law  of  Exponents). 

—  8  ct^ic*  aj2  .  a;4  —  3j2+4     _  y^  (Law  of  Exponents). 

Hence,  the  product  is  —  8  a^^e. 
Multiply : 

2.  10  a^  3.   -5mV  4.   -4a&c  5.        '^cC-bc^ 

5  o?  3  mn  2  a-b  —  7  a6^c 

6.  x'f  7.       5i9g2a^  8.   -Sab 

xy^  —  2  rcfx  —  1 

10.   —2  a?  11.   —  6a-c-.T         12.   —  3a5 

2a^  -4a367i  2  6a 

16.    -1 

-1 

20.    —y\ 

ap'-q^ 


14. 

-3n« 
6  5« 

18. 

4a2 
-1 

22. 

2a-+^ 
3a2 

26. 

52/ 

30. 

—  5  a;'* 
a; 

34. 

—  a"' 

—  a'* 

38. 

7. 

5pq^x^ 
-2rq'x 

11. 

—  6  a-c'x 

—  4  a^bn 

15. 

4  a^^y 
3  a352^ 

19. 

10  m*n' 

23. 

-2a^b'^ 
7  a^'-6*^ 

27.  a^b^a^y""-'^ 

31.    — a;?/2; 

35.    —  4a^&'' 
-3a'-^6^ 

39.  8?-==-^ 
3  r^s«-2^ 


24. 

—  x^y"" 

xy 

28. 

32. 

ab'(^ 
d^b'^-c 

36. 

yn-m 
ym-n+1 

40. 

z" 

9. 

-2aa^ 

13. 

-2aV 
-4aa;* 

17. 

—  5  m^d^ 

-2m''cd' 

21. 

-2ahnhi' 
Sb'nV 

25. 
29. 

Ax"-' 

-2af'+i 

33. 

2^V-« 

2n-3^2a-6 

37. 

—  xy 

41. 

mTiffb-y^ 

MULTIPLICATION  55 

90.  Product  of  several  monomials. 

By  the  law  of  signs,     —a-  —  b=  -\-cib', 

—  a'—b'  —  c=-{-cUt'  —c——  abc ; 
—  U'—b'—C'—  (1=—  abc  •  —  d  =  -f  abed ;  etc. 

The  product  of  an  even  number  of  negative  factors  is  positive  ; 
of  an  odd  number  of  negative  factors,  negative. 

Positive  factors  do  not  affect  the  sign  of  the  product  (§  83). 

EXERCISES 

91.  Find  the  products  indicated : 

1.  (_1)(-1)(-1)(-1).        4.  (-2xy)(-8xy)(5x^y)(-xf). 

2.  (-2)(-ab)(-Sa'').  5.  (-Uc)b(-3c')c(-b)(-bc). 

3.  (-a''x)(4bx)(-5a^.         6.  (-2»)(-2^)(5  •  2»)(-5*.2). 

7.    Find  the  product  of  2",  2'-',  and  2"+^     Test  the  correct- 
ness of  your  answer  when  n  represents  3. 

92.  To  multiply  a  polynomial  by  a  monomial. 

4.3  40  +  3  4^-1-3  5a -26 

9  9  9  5« 


86  80  4-6  8<-h6  15a -66 

In  general,  a(x  +  y  -\-z)  =  ax -f- ay  -f  «2.     That  is, 

93.  Tlie  product  of  a  polynomicd  by  a  monomial  is  equal  to  the 
algebraic  sum  of  the  partial  products  obtained  by  multiplying 
<ich  term  of  tht  itohjnominJ  by  the  monomial. 

This  is  the  distributive  law  for  nmlti|)lication. 


EXERCISES 

94.   Multiply: 

1.  3ar'-2a7/by  5a^.  A.  i^<f  - 2 p(f  hy  - pq, 

2.  3  a'*  -  6  ii%  by  -  2  6.  5.  4  a^  -  5  bh  -  c"  by  ahc\ 

3.  m-n^  —  .S  mn^  by  2  mn,  6.  —  2 ac  -|-  4  ax  by  —  5  acx. 


56  MULTIPLICATION 

Perform  the  multiplications  indicated  : 

7.  a'bc (3  a^  -  4  a^b  -  5  a'b^  +  2  air'  -  16  b*). 

8.  2  a;?/(5  ;«2  - 10  iP2/  -  36  /  -  5  a;  +  5  2/  + 120). 

9.  5  m^(16  »r  -  20  m*/i + 13  mn^  -25n% 

10.  a6c(a-/>-  -  2  a'c'  -  2  6V  -  a'  -4:b'-c'-  5  a6c). 

11.  -  bc{b*  +  c''  -  ¥  -  c^  +  6V  _  4  ^2^  _p  8  6(r^  -  2  6c). 

12.  m"ii'^(^)i*  —  5  ?)i^?<,*  —  16  m^n^  +  24  mri*  —  w**). 

13.  x''-^y"'-^\a^y'"~^  —  5  a^^-"?/'"-^  _|_  IQ  ^-nym-i  _  5  ^4-2,y-m  _^ 

95.   To  multiply  a  polynomial  by  a  polynomial. 

EXERCISES 

1.  Multiply  a;  +  5  by  X  -f  2;  test  the  result. 

PROCESS  TEST 

a?  +  5  =6  when  x  =  l 

X  ^2  =_3 

ic  times  (ic  +  5)  =  a^  -h  5  ic 
2  times  (ajH-5)=  2a;+10 

{x  -f-  2)  times  (a;  +  5)  =  a;^  +  7  aj  + 10         =18 

Test.  —  The  product  must  equal  the  multiplicand  multiplied  by  the 
multiplier,  regardless  of  what  value  x  may  represent.  To  test  the  result, 
therefore,  we  may  assign  to  x  any  value  we  choose  and  observe  whether, 
for  that  value,  product  obtained  =  multiplicand  x  multiplier.  When 
X  =  1,  multiplicand  =  6,  multiplier  =  3,  and  od^  +  1  x  +  \^  =  \d>  \  since 
6  X  3  =  18,  it  may  be  assumed  that  x^  +  7  x  +  10  is  the  correct  product. 

EuLE.  —  Multiply  the  multiplicand  by  each  term  of  the  multi- 
plier arid  find  the  algebraic  sum  of  the  partial  products. 

2.  Multiply  a?  +  4  by  aj  +  6 ;  test  the  result  when  a.-  =  1. 

3.  Multiply  a^  —  1  by  .^•  —  2 ;  test  the  result  when  x  =  5. 

4.  Multiply  2x-\-3  by  4a;  —  1;  test  the  result  when  x=l. 

5.  Multiply  a;^  +  a;  H- 1  by  a;  —  1 ;  test  the  result  when  x  =  2. 


MULTIPLICATION  57 

6.  Multiply  2a  —  6  +  cby  3a  +  6;  test  the  result  when 
0  =  1,  6  =  1,  and  c  =  l. 

In  like  manner  the  multiplication  of  any  two  literal  expressions  may 
be  tested  arithmetically  by  assigninu;  any  values  we  please  to  the  letters. 

While  it  is  usually  most  convenient  to  substitute  +  1  for  each  letter, 
since  this  may  be  done  readily  by  adding  the  numerical  coefficients,  the 
student  should  bear  in  mind  that  this  really  tests  the  coefficients  and  not 
necessarily  the  exponents,  for  any  power  of  1  is  L 

Multiply,  and  test  each  result : 

7.  2a;  +  3by  x-t-2.  12.  4y-6bhy2y  +  b. 

8.  ix  +  lhy  3x-\-A.  13.  2b-\-5chy  5b-2c. 

9.  5  n  -  1  by  4  ?i  -h  5.  14.  ab  -Why  ab-^  10. 

10.  h-i-2k  hy  3 h  —  k.  15.   ax -^ by  hy  ax— by. 

11.  3l-\-5thy  2l-\-(yt.  16.    a^  — ay -^f  hy  a -\-y. 

An  indicated  product  is  said  to  be  expanded  when  the  mul- 
tiplication is  performed. 

Expand,  and  test  each  result : 

17.  (x-\-y)(x-\-y),  22.  (x- ^  y-)(af  ^  y^y 

18.  (e^  +  d«)(e'-i-rf^).  23.  (xr  +  y'')(3if'-y^), 

19.  (3a4-6)(3a  +  6).  24.  (3  ax +  2  by) (3  ax -{-2  by), 

20.  (3a  +  b)(3a-b),  26.  (3ax-\-2by)(3ax-2by). 

21.  {2a^  +  b)(2a'^-b).  26.  (a -{- b  +  c){a -[- b  -  c). 

Multiply,  and  test  each  result  : 

27.  2a''-3h'-abhy3a^-4alf-5b'. 

28.  r>x-5x'-{-10hyl2-30x-]-2x'. 

29.  3  n*  -I-  3  m*  4-  7nn  hy  m^  —  2  mn^  -f  mhi. 

30.  4/r'-10-f2yby  2/-3.v-f  r>. 

31.  4a;-3ar*-f-2ar'by  3a;-10jr^-f  10. 


58  MULTIPLICATION 

Multiply,  and  test  each  result : 

32.  a'  +  a*-^4ta^-a^  +  ahy  a-\-l. 

33.  31a^-27a;^  +  9a;-3by  3a;  +  l. 

34.  4rx^~3x'y  +  5xf'-6fhy5x  +  6y. 

35.  a-\-b-\-c-\-dhja  —  b  —  c-\-d. 

36.  a^  -f  6^  +  c^  —  a6  —  ac  —  6c  by  a  +  6  4-  c. 

37.  m^  —  m^n^  +  m^n^  —  mhi^  +  n^  by  m^  -f  w^. 

Expand : 

38.  (|.a+i6)(.i.a-i6).  41.  (i  6'^-i  6  +  i)a  6- i). 

39.  (|a^  +  j2/)(i^  +  i2/).  42.  (f  0^-0^2/  + 1 2/^(1  a? +  1  y). 

40.  (^m-i7i)(fm-f7i).  43.  (|  r^^-i  ^^^ +  f  n- i)(3  n+4). 

44.  (a  +  6)(a-6)(a  +  6)(a-6). 

45.  (a-6)(a  +  6)(a2  +  62)(ti4^54). 

46.  (2a  +  36H-5c)(2a  +  36-5c). 

47.  (5  m  —  2  w  +  a;)(5  7?i  —  2  n  —  a;) . 

48.  (a'"-6«)(a'"  +  5'<)(a2'«  +  52n-^. 

49.  {x""  —  riaj"-^2/  +  i  nx''-^y^)(x  4-  y). 
Multiply : 

50.  aCC^n  _|_  ^^2«  ]^y.  ^r^n  _  ^y2n^ 

51.  a.'c'*-^  +  ^""^  by  3  aa;"-i  +  2  2/'^-^ 

52.  a.-2"  +  2  a;"?/'*  +  /"  by  a;-"  —  2  a;"?/"  +7/-". 

53.  a^  +  a''*62^  +  a^"^^-  _|_  56c  ]^y  ^2n  _  520^ 

54.  m^'+V-i  +  m^-'?i^+^  + 1  by  m'-hi'+^  —  ?>i*+^n'-i  + 1. 


55. 


X.  z'ia+1  _X^2a_^^  ^2a-l  ^y  2  ;22«-l  +  2  ^^a-S  _^  2  ;32«-3. 


96.  When  a  polynomial  is  arranged  so  that  in  passing  from 
left  to  right  the  several  powers  of  some  letter  are  successively 
higher  or  loiver,  the  polynomial  is  said  to  be  arranged  according  to 
the  ascending  or  descending  powers,  respectively,  of  that  letter. 

The  polynomial  x'^  —  4  x^y  -{- 6  x^y^  —  4xy^  +  ?/*  is  arranged  according 
to  the  ascending  powers  of  y  and  the  descending  powers  of  x. 


ML  LII  PLICATION  59 

97.  When  polynomials  are  arranged  according  to  the  ascend- 
ing^' or  the  descending  })owers  of  some  letter,  i)rocesses  may 
often  be  abridged  by  using  the  detached  coefficients. 

BXBRCISBS 

98.  1.    Expand(2a!*-3.r''4-.3.r-hl)(3a;4-2). 

FULL    PROCESS  DETACHED    COEFFICIENTS 

'j,*-3a^-\-Sx+l  2     -3     +0     +3     +1 

±2 3     +2 

-9aj<  -\-9a^-\-3x  6     -9     +0     -f9     +3 

4a?*~6ar^  4-6x4-2  4     -6     4-0     +6    H-2 

.i.r-5a;*-6ar»+9a^  +  9a:-f  2  ^,t^-o  x^ -60^ +  9a^ +  9  x-\-2 

Since  the  second  power  of  x  is  wanting  in  the  first  factor,  the  term,  if 
it  were  supplied,  would  be  Ox'^.  Therefore  the  detached  coefficient  of 
thf  term  is  0.  The  detached  coefficients  of  missing  terms  should  be  sup- 
plied to  prevent  confusion  in  placing  the  coefficients  in  the  partial  products 
and  to  avoid  errors  in  writing  the  letters  in  the  result. 

Arrange  properly  and  expand,  using  detached  coefficients: 

2.  (a;-f  a5»4-l4-a:^(a;-l). 

3.  (x»4-10-7a;-4a^(x-2). 

4.  (U-^9x-6x^  +  x')(x  +  l). 

5.  (a»-|-4a2-lla-30)(a-l). 

6.  {-la'-2a^-Sa  +  a*-3){2  +  a). 

7.  (2m-34-2m»~4m*)(2m-3). 

8.  {x-\-x'-o)(x^-3-2x), 

9.  (6»  +  r)6-4)(-44-262-36). 

10.  (4  11^  4-  r>  -  2  ?i*  -f  16  71  -  8  w^  _|_  n')(n  4-  2). 

11.  :   -6a;-f5«*-4a^4-3a^-2a:'4-<)(a^-f 2x4-1). 

12.  (1  -f  .r  4-  4  .T^  -}-  10  .r'  +  40  r'  4-  22  x*)(2  x^  +  l-3x). 


60  MULTIPLICATION 

99.  Multiply  a^  +  2  aV)  -h  2  ah''  +  W  by  a^  +  ah-{-  h\ 

PROCESS  TEST 

a«  +  2a26  +  2a6-  +6'  =    6 

or+    ah  +h^  =3 

a^  +  2a^6  +  2a«62+    a'h^ 

a^h  +  2a^W-\-2a'W+    ah' 

a^h^  +  2a'W  +  2ah'  +  h' 
a^  +  3  a^6  +  5  a^h-  +  5  o^^'^  +  3  a6^  +  &'  =18 

When  each  letter  of  an  expression  is  given  the  value  1,  the 
expression  is  equal  to  the  sum  of  its  numerical  coefficients. 
The  test  on  the  right  of  the  process,  then,  tests  the  signs  and 
coefficients  in  the  product,  but  not  the  exponents. 

To  test  the  exponents  in  the  product,  observe  that  each  term 
of  the  multiplicand  contains  three  literal  factors,  as  ooa,  aah, 
etc.,  or  is  of  the  third  degree;  also  that  each  term  of  the  mul- 
tiplier is  of  the  second  degree.  Therefore,  each  term  of  the 
product  should  be  of  \hQ  fifth  degree. 

When  all  the  terms  of  an  expression  are  of  the  same  degree, 
the  expression  is  called  a  homogeneous  expression. 

The  multiplicand  in  the  process  is  a  homogeneous  expression  of  the 
third  degree  ;  the  multiplier  is  a  homogeneous  expression  of  the  second 
degree  ;   and  the  product  is  a  homogeneous  expression  of  the  fifth  degree. 

As  a  further  test  observe  that  the  multiplicand  involves  a 
and  h  in  exactly  the  same  way,  h  corresponding  to  a,  h^  to  a^, 
and  b^  to  a^,  so  that  if  h  and  a  were  interchanged  the  multi- 
plicand would  not  be  changed,  except  in  the  order  of  terms. 
Such  an  expression  is  said  to  be  symmetrical  in  a  and  h. 
Since  both  multiplicand  and  multiplier  are  symmetrical  in  a 
and  h,  the  product  should  be  symmetrical  in  a  and  b. 

100.  TJie  product  of  tivo  homogeneous  expressions  is  a  hoino- 
geneous  expression. 

If  two  expressions  are  symmetrical  in  the  same  letters,  their 
product  is  symmetrical  in  those  letters. 


MULTIPLICATION  61 

EXERCISES 

101.  I^xpaud,  and  test  each  result: 

1.  {a-\-b-\-c)(a-{-b  +  c). 

2.  («-  -  <ih  4-  b'^){a-  -\-ab-i-  b^. 

3.  (a^-\-3a-b-\-3ab*-^t/^(a-\-b). 

5.  (a^-\-b''-\-c'-\-d')(a^-b^-\-c'-ci:^. 

6.  {ab  +  bc  +  cd  +  bd){ab  -\-bc  —  cd  —  bd). 

7.  {s?-xy  +  f  +  x+y-\-l){x^y-^l), 

8.  (a»  +  3a26-h3a5«H-63)(a«-|-2a6  +  6*). 

9.  (a*  — a6-ac  +  6*  — 6c  +  c2)(a4-6-hc). 

Addition,  Subtraction,  Multiplication 

EXERCISES 

102.  1.    Simplify  a- 4- a(6- a)- 6(26 -3  a). 

Solution.  —  The  expression  indicates  the  algebraic  sum  of  a^,  a(b  —  a)^ 
and  —  6(2  6  -  3a).  Expanding,  a{h  —  a)=ab-  a^,  and  -  6(2  6  -  Sa) 
=  -  2  6-2  +  3  ah.  Therefore,  writing  tlie  terms  in  order  with  their  proper 
signs,  a^  +  a(b  -  a)  -  6(2  6  -  3  a)  =  a*-^  +  a6  -  a^  -  2  6^  +  3  «6  =  4  a6 -2  b^. 

Simplify : 

2.  x'-{-x(y-x).  6.  x^-f-ix-yf. 

3.  c-  —  c{c  —  d).  6.  c(tt  — 6)  — c(a4-6). 

4.  5-2(a;-3).  7.  a^- 63-3a6(a-6). 

8.  -2{3^-xf)-b{xf-x'y). 

9.  (3a-2)(2o-3)-6(a-2)(a-l). 

10.  8r»-(4a;*-2a^  +  y*)(2aJ  +  y). 

11.  (3  m  -  1  )(m  +  2)  -  3  m{m  +  3)  4-  2(  ///  -f- 1). 

12.  (a-6)--2(a2-6-)-2a(-f/      A,      \  h\ 


62  MULTIPLICATION 

13.  4  (ax  —  bx  -\-cx  —  dx)  —  3(ax  -j-  hx  —  ex  —  dx). 

14.  (x  +  l)(x  +  2)  -  2(x  -  l)(x  -  2)  -h  4.(x  +  2)(x  +  3). 

15.  (x'  +  2xy-\-  f)(x'  -  2  xy  +  /)  -  (ar'  +  f)(x^  +  f), 

16.  b*  +  («'  -  «&  4-  'J')(a2  +  b'')  -  (a^  -  b^)(a  +  2  b). 

17.  7/3-[2a;«-a;7/(a;-2/)-2/«]+2(a;-2/)(x2^^y_^^2^^ 

103.    Numerical  substitution. 

EXERCISES 

1.  When  a  =  —  2,  6  =  3,  c  =  4,  find  the  value  of 

a2-(a-c)(6  +  c)  +  2  6. 

SoLUTiox.      a2- (a-c)(6  +  c) +  2  6=  (-2)2- (-2-4)(3-f  4) +2.  3 

=  4-(-6)(7)  +  6 
=:4-(-42)  +6 
=  4  +  42  +  6  =  52. 

When  x  =  3,  y=  —4t,  z  =  0,  m  =  6,  n  =  2,  find  the  value  of : 

2.  7n(x  —  y)-^z'^.  5.    (x-\-y){m  —  7i)  +Sz. 

3.  z  +  m'-iy^-l).  6.    (m  +  a;)^- (m -2/)2-/. 

4.  x^  —  y^  —  711-  ■i-7r.  7.    a7?/2;  —  ?i  (cc  —  m) ^  —  (/la;) ^ 

8.  3  m(??i  —  w)  +  4  71(2/  —  a;)  —  7(2/  +  2). 

9.  ^(y  -2  71)  -  ^(71-2  y)(S  y  -An), 

10.  x'y-(7n  —  7iy(m  -\-n)  -\-  (7n  +  ?i)^(m  —  n). 

11.  (a;-?/)2-x2/(aj-2/)(a;4-2/)(x-2  +  2/^). 

12.  3  7n(x  —  y  —  7if  —  (y  —  n  —  x)(7i  —  x  —  y). 

13.  (2  a?  +  2/)"  -  (^  -  2  2/)"  -  (m  +  n)2(a;  +  2/  +  2)^ 

14.  (x  -J-  2/  +  2)2  —  ar2/(2/  +  2;  —  .t)  (a;  +  2:  —  2/)  —  2;(a;  +  2/  —  ^)' 

15.  (??i  +  n  +  aj)**  —  (m  -f-  n  —  xy  —  {7n  —  71 -\-  a;)"(  —  m  +  n  +  a;)''. 

16.  Show  that  (a-b  +  cy  =  a'  +  b^-\-  c^  -2  ab  -h2  ac-2bc, 
when  a  =  1,  5  =  2,  and  c  =  3 ;  when  a  =  4,  6  =  2,  and  c  =  —  1. 

17.  By  substituting   numbers   for  a,   b,  and  c,  show  that 
(a  -\-b)(bi-  c)(c  -h  a)  -\- abc  =  (a  +  b  +  c){ab,  +  6c  +  «c). 


MULTIPLICATION  63 

SPECIAL   CASES    IN   MULTIPLICATION 

104.  The  square  of  the  sum  of  two  numbers. 

Sliuw  by  actual  multiplication  that 

(a  +  &)(a -f  6)  =  a* -I- 2  a6  4- 6*; 
also  tliat  (x  +  y){x  -\-y)^x^  ■\-2xy +  f. 

105.  Principle.  —  The  square  of  the  sum  of  two  numbers  is 
e<nial  to  the  square  of  the  first  number,  i^lus  twice  the  product  of 
the  first  and  second,  plus  the  square  of  the  second. 

Since  5  a*  X  5  a'  =  25  a«,  3  a%'  x  3  a^l^  =  9  aV,  etc.,  it  is  evi- 
dent that  in  squaring  a  number  the  exponents  of  literal  factors 
are  doubled. 

EXERCISES 

106.  Expand  by  inspection,  and  test  each  result: 

1-  {p  +  ^){P^<i)'  8.  {^x  +  z)\  15.  {a'  +  by. 

2.  (r-f  .s)(r  +  s).  9.  (2a-\-xy.  16.  {(t^-^Wf. 

3.  {a-\-x){a  +  x).  10.  {nb-\-cdf.  17.  (a"H-^")2. 

4.  (x-f4)(a;  +  4).  11.  {^x  +  2y)\  18.  (a:-4-2/")^ 

5.  (a4-6)(a  +  6).  12.  (7z  +  3c)l  19.  Q^a^  +  ol^f. 
6-  ill  4-  T)(y  -h  7).  13.  (3  6  + 10  x)\  20.  (1-1-2  a^bf. 

7.    (2-|-l)(z-|-l).        14.    (3a^-h4ar*2/)*.      21.    (x" -  ^ -h  t/**  ~  ^ . 

107.  The  square  of  the  difference  of  two  numbers. 
Show  by  actual  multiplication  that 

(a_6)(a-6)  =  a*-2o6-h&*; 
'M)that  (x-y)(x-y)  =  x'-2xy-\-f. 

108.  Principle.  —  Tlie  square  of  the  difference  of  two  num- 
h.  rs  is  equal  to  the  square  of  the  first  number,  minus  twice  the 
I'loduct  of  the  first  and  second,  plus  the  square  of  the  second. 


64  MULTIPLICATION 

EXERCISES 

109.  Expand  by  inspection,  and  test  each  result : 

1.  (x—m)(x—7n).  9.  (2a— xf.  17.  (3a;  — 2)2. 

2.  {m—n){m—n).  10.  (S^n  —  ny.  18.  (2x  —  5yy. 

3.  (x-6)(x-6).  11.  (4:x-yy.  19.  (om-Sny 

4.  (p-S)(p-S).  12.  (m-4w)l  20.  (3p-5  9)2. 

5.  (g_7)(g-7).  13.  (p-3qy  21.  (a»-6")2. 

6.  (a-c)(a-c).  14.  (a -7  by.  22.  (x'^-y^^y. 

7.  (a-a;)(a-.T).  15.  (4 a -3)2.  23.  (a;'"-i  -  i/"-^)^. 

8.  {x—l)(x--l).  16.  (5a;  — 4)2.  24.  {mx^^  —  ny'^y. 

110.  The  square  of  any  polynomial. 

Show  by  actual  multiplication  that 

{a  +  h  +  cy=  a^  ^h^  +  c"  +  2  ab  +  2  ac  +  2hc', 
al so  that       {a  +  h  +  c  +  dy  =  a"  +  Jy^ -^ c^  +  dP -\-2 ah  -^2ac 

+  2ad^2hc-\-2hcl^2cd. 

Similarly,  by  squaring  any  polynomial  by  multiplication,  it 
will  be  observed  that : 

111.  Principle. —  Tlie  square  of  a  polynomial  is  equal  to  the 
sum  of  the  squares  of  the  terms  and,  twice  the  product  of  each  term 
by  each  term,  taken  separately,  that  folloivs  it. 

When  some  of  the  terms  are  negative,  some  of  the  double  products  will 
be  negative,  but  the  squares  will  always  be  positive.  For  example,  since 
(-  6)2  =  +  62^  (a  -  6  +  c)2  =  a2  _,.  (_  5)2  4.  c2+2 a(-6)  +2  ac+2(-&)c 
=  a2  +  62  _|_  c2  _  2  a6  +  2  ac  -  2  6c. 

EXERCISES 

112.  Expand  by  inspection,  and  test  each  result : 

1.  (x  +  y  +  zy.  3.    (x-y-zy.  5.    (x^y-^zy. 

2.  (x  +  y  —  zy.  4.    {x-y  +  zy.  6.    (x  —  y  +  3zy. 


MULTIPLICATION  65 

Expand  by  inspection,  and  test  each  result : 

7.  (a-26  +  c)-.  13.    {Sx-2y  +  4zy. 

8.  (2a-6-cf.  14.    (2o-56-h3c/. 

9.  (b-'Ja-tc)'.  15.    (2m-^n-ry. 

10.  (ax-by-\-czy.  16.    (12 -2 x-^ 3 yf. 

11.  (qb-pc-rdy.  17.    (a  +  m -f  6  +  w)*. 

12.  (dc  —  bd  —  dey.  18.    (a  — m-H&  — w)^. 

113.  The  product  of  the  sum  and  difference  of  two  numbers. 

Let  a  and  b  represent  any  two  numbers,  ci  +  ^  their  sum, 
niid  a  —  b  their  difference. 

^iiow  by  actual  multiplication  that 

(a-^b){a-b)  =  a^-b\ 

114.  PRrxciPLE.  —  The  product  of  the  sum  and  difference  of 
two  numbers  is  equal  to  the  difference  of  their  squares. 

EXERCISES 

115.  Expand  by  inspection,  and  test  each  result : 

1.  (x  +  y)(x-y),  11.  {ab  +  5)(ab-5). 

2.  (a-{-c)(a-c).  12.  (cd-f  d«)(cd-d*). 

3.  (y>  +  7)(p-7)-  13.  (2x  +  3y)(2x-3y). 

4.  (p^5)(p-5).  14.  (3wH-4?i)(3m-4ri). 

5.  (a;4-l)(x-l).  15.  (12  +  xy)(12  -  xy). 

6.  (x»  +  l)(ar*-l).  16.  (3mhi-b)(3  mhi +  b). 

7.  (a^4-l)(ar»-l).  17.  (06  + cd)(a6  -  cd). 

8.  (j^-l)(a:*  +  l).  18.  (2  a:»-h5  2/')(2  x^-Sy^. 

9.  (a-^_l)(a:»4-l).  19.  (3  a« -h  2  ^^(3  a^  -  2 /). 

10.    (j--f  y3)(a;-_7/3).  20.    (2  a' +  2  b%2  a^ -2)  b^. 

milnk'8  stand.  ALG,  — .J 


66  MULTIPLICATION 

21.  (—5n  —  b){—5n-\-b).  24.    (a;"-^ -h2/''"^^)(a;"»-*  -  2/"+*). 

22.  (-x-2y)(-x  +  2y).  25.    (3  af +  7  2/'*)(3  af*- 7  2/"). 

23.  (-4-3fl,)(-4-3a).         26.    (5  a362_|.2  af)(5  ^362-2  a."). 

One  or  both  numbers  may  consist  of  more  than  one  term. 

27.  Expand  (a-\-7n  —  n){a  —  m-\-n). 

Solution 
a  +  wi  —  n  =  a  +  (m  —  w). 
a  —  m  +  n  =  a  —  (wi  —  n). 
.-.  [a  +  wi  —  n][a  —  m  +  «]  =  [«+  (w—  n)][a  —  (rn  —  w)] 
Prin.,  §  114,  =  a^  -  (m  -  7i)2 

§  108,  =  a2  _  (wi2  _  2  wri  +  n^) 

=  a^  —  m^  +  2  mn  —  n^. 
Expand : 

28.  {a-^x-y){a-x  +  y).  33.  {y  ^  C  + d){y -^c-d). 

29.  (a;  +  c  — f^)(x  — c  +  cZ).  34.  ia-\-x-\-y){a-{- x  —  y). 

30.  (r+p-^)(r-p  +  g).  35.  (x2  + 2a;4- 1)  (a^  +  2  a;-l). 

31.  {r+p^q){r-p-q).  36.  (a;^  _|_  2  a;- l)(a^- 2  a;  +  l). 

32.  {x-irh  +  n){x-h-n).  37.  (a^  +  3  a;  -  2)(ar^-3  a;  +  2). 

38.  (m*-2m2+l)(m^  +  2m2  +  l). 

39.  (2a;  +  32/-42j)(2a;  +  32/  +  42). 

40.  (2x'-xy-^'dy^){2x'  +  xy-Zy'^. 

41.  (x^-^xy-\-y'^){:t?  —  xy-\-y'^). 

42.  [(a  +  6)  +  (c  +  d)]  [(a  +  6)  -  (c  +  d)]. 

43.  (a  +  6  +  a;4-2/)(«+^  — ^-2/)- 

44.  {a-\-h-\-m—n){a-\-h  —  m-\-n). 

45.  (a;  — m  +  2/  — ^^)(a:  — wl--2/  +  ^)• 
46.    {p  —  q-\-r-\-s){p  —  q  —  r  —  s). 
47.    (a  — m  — 6  — n)(a  +  m  — 6  +  w). 


MULTIPLICATION  67 

116.   The  product  of  two  binomials  that  have  a  common  term. 
Let  x-^a  and  x  i-b  represent  any  two  binomials  having  a 
common  term,  x.     Multiplying  x  i- a  by  x -\- b, 

x-\-a 
x-\-b 
a^-^-ax 

bx  +  ab 


x^  +  {a-{-b)x-\-ab 

117.  Principle.  —  The  product  of  tico  binomials  having  a 
common  term  is  equal  to  the  sum  of  the  sqitare  of  the  common 
term^  the  product  of  the  sum  of  the  unlike  terms  and  the  common 
term,  and  the  product  of  the  unlike  terms. 

EXERCISES 

118.  1.    Expand  {x  -\-2){x  +  5)  and  test  the  result. 

Solution.  —  The  square  of  the  common  term  is  x"^ ; 

the  sum  of  2  and  5  is  7  ; 

the  product  of  2  and  5  is  10  ; 

.-.  (a;  +  2)  (x  +  6)  =  a;*^  +  7  «  +  10. 
Test.  —  If  x  =  1,  we  have  3  •  6  =  1  +  7  +  10,  or  18  =  18. 

2.  Expand  (a  -f  1)  (a  —  4)  and  test  the  result. 

Solution.  —  The  square  of  the  common  term  is  a^  ; 
the  sum  of  1  and  —  4  is  —  3  ; 
the  product  of  1  and  —  4  is  —  4  ; 
.-.  (a  +  1)  (a  -  4)  =  a--*  -  3  a  -  4. 
Test.  —  If  a  =  4,  we  have  6  •  0  =  16  —  12  -  4,  or  0  =  0. 

3.  Expand  (n  —  2){n  —  3)  and  test  the  result. 

Solution.  —  The  square  of  the  common  term  is  n^ ; 

the  sum  of  —  2  and  —  3  is  -  5  ; 

the  product  of  —  2  and  —  3  is  6  ; 
.-.  (n  -  2)(n  -  3)  =n^-bn  +  Q. 
Test.  —  If  n  =  3,  we  have  1  •  0  =  9  —  16  -f  0,  or  0  =  0. 


68 


MULTIPLICATION 


Expand  by  insi)ection,  and  test  each  result 


4. 

(a;  +  5)(x--f6). 

5. 

(x  +  7)(x-\-S). 

6. 

{x-7)(x  +  H), 

7. 

(^  +  7)(.T-8). 

8. 

(x-5)(x-4.). 

9. 

(x-3){x-2). 

10. 

(:x-5)ix-l). 

11. 

(x-{-o)(x-\-H). 

12. 

(p_4)(p  +  l). 

13. 

(r-3)(r-l). 

14. 

(n-8)(n-12). 

15. 

(n-6)(n-\-15). 

16. 

{a^  +  5)(x^-3). 

17. 

[x^-7)(x'  +  6). 

18.  (a;"-5)(iK''  +  4). 

19.  (af  -  a)(x'' -  b). 

20.  (y-2a)(y-^3b). 

21.  (z  — 4a)(2!  +  3a). 

22.  (2x  +  5){2x-\-S). 

23.  (2a;-7)(2a;  +  5). 

24.  (3y-l)(3y  +  2).    . 

25.  (4a;2  +  l)(4a!2_7). 

26.  (a6-6)(a?>+4). 

27.  (a^/-a)(a;y  +  2a). 

28.  (3xy  +  y^)(y^  —  xy). 


29.    (x'  +  .y-l)(a;  +  z/  +  2). 


30.    (x-2/-2)(ic-2/-8). 


31.    {x'-  +  x-l)(x^  +  x-h3). 

By  an  extension  of  the  method  given  above,  the  product  of 
any  two  binomials  having  similar  terms  may  be  written. 


32.    Expand  (2  a!  -  5) (3  x  + 4) 

PROCESS 

2x-5 

X 

8a;  +  4 


Explanation.  —  The  product  must  have  a  term 
in  x^,  a  term  in  x,  and  a  numerical,  or  absolute,  term. 

The  x^  term  is  the  product  of  2  x  and  3  a;;  the  x  term 
is  the  sum  of  the  partial  products  —  5  .  3x  and  2  x  •  4, 
called  the  cross-products  ;   and  the  absolute  term  is  the 


6  .T^  —  7  «  —  20     product  of  -  5  and  4 

The  process  should  not  be  used  except  as  an  aid  in  explanation. 

Expand  by  inspection,  and  test  each  result : 

33.  (2a;-f-5)(3a;  +  4).  36.    (3x-y)(x -3y). 

34.  (3a;-2)(2ic-3).  37.    (2a-f  5  6)(5«4-2  6). 

35.  (3a-4)(4a  +  3).  38.    (7  n' -2p)(2n^ -7p). 


MULTIPLICATION  69 

Algebraic  Representation 

119.  1.  Express  in  the  shortest  way  the  sum  of  five  x'a ; 
the  product  of  five  x's. 

2.  When  the  multiplicand  is  x  and  the  multiplier  y,  express 
the  product  in  three  ways. 

3.  Indicate  tlie  product  when  the  sum  of  x,  y,  and  —  (Z  is 
multiplied  by  xy. 

4.  How  much  will  a  man  whose  wages  are  a  dollars  per 
•lay  earn  in  b  days  ?  in  c  days  ?  in  x  days  ?  in  a  days  ? 

5.  If  a  man  earns  a  dollars  per  month  and  his  expenses  are 
b  dollars  per  month,  how  much  will  he  save  in  a  year  ? 

6.  Indicate  the  sum  of  x  and  z  multiplied  by  m  times  the 
sum  of  X  and  y. 

7.  From   x  subtract  in   times  the  sum   of  the  squares  of 
(a  -f  b)  and  (a  -  b). 

8.  A  number  x  is  equal  to  (y  —  c)  times  {d-\-c).     Write 
the  equation. 

9.  How  many  seconds  are  x  days  -|-  c  hours  +  d  minutes  ? 

10.  Express  in  cents  the  interest  on  y  dollars  for  x  years,  if 
the  interest  for  one  month  is  z  cents  on  one  dollar. 

11.  How  far  can  a  wheelman  ride  in  a  hours  at  the  rate  of 
fi  miles  an  hour  ?  How  far  will  he  have  ridden  after  a  hours, 
if  he  stops  c  hours  of  the  time  to  rest? 

12.  How  many  square  rods  are  there  in  a  square  field  one 
of  whose  sides  is  2  b  rods  long  ?  (x  —  y)  rods  long  ? 

13.  What  is  the  number  of  square  rods  in  a  rectangular 
field  whose  length  is  (a  +  b)  rods  and  width  (a  —  b)  rods  ? 

14.  A  fence  is  built  across  a  rectangular  field  so  as  to  make 
the  part  on  one  side  of  the  fence  a  square.  If  the  field  is  a 
rods  long  and  b  rods  wide,  what  is  the  area  of  each  part  ? 

15.  Represent  (a  —  b)  times  the  number  whose  tens'  digit 
is  X  and  units'  digit  y. 


70  MULTIPLICATION 

Equations  and  Problems 

120.     1.   Given  5(2  x  -  3)-  7(3  .t  -h  5)  =  -  72,  to  find  the 
value,  of  X. 

Solution 

5(2  a:  -  3)  -  7(3  x  +  5)  =  -  72. 
Expanding,  10  x  -  15  —  21  x  —  35  =  -  72. 

Transposing,  10  x  -  21  x  =  15  +  35  -  72. 

Uniting  terms,  —  11  x  =  —  22. 

Multiplying  by  -  1,  11  x  =  22. 

.-.  x=2. 

Verification.  —  Substituting  2  for  x  in  the  given  equation, 
5(4-3)  -7(6  +  5)  =-72. 
5 -77  =-72. 
Hence,  2  is  a  true  value  of  x. 

Find  the  value  of  x,  and  verify  the  result,  in : 

2.  2  =  2x-5-(x-S).  4.   l  =  5(2a;-4)  +  5a;-h6. 

3.  10a3-2(a;-3)  =  22.  5.    7(5-3a;)  =3(3-40;) -1. 

6.  2(a;-5)4-7=^  +  30-9(a;-3). 

7.  5-\-7(x-5)  =  15(x  +  2-S6). 

8.  (a;-2)(a;-2)  =  (a;-3)(.'»-3)  +  7. 

9.  (x-4)(x  +  ^)  =  (x-6)(x-\-5)-\-25. 

10.  Ax^-A(a^-x'  +  x-2)  =  Ax', 

11.  7(2.T-36)  =  26-3(2a;  +  6). 

12.  S{2b-Ax)-{x-b)  =  -6b.       14.   3(a;-a- 25)  =  36. 

13.  4:x-x'  =  x(2-x)  +  2a.  15.    5b  =  3(2 x-b)-4:b. 

16.  x'-(2x-\-S)(2x-'S)-\-{2x-Sy  =  (x  +  9)(x-2)-2. 

17.  3(4 -0^)2-2(0^  +  3)  =  (2 a;- 3)2- (a;  +  2)(a;- 2) +  1. 

18.  20(2-x)  +  3(a;-7)-2[o;  +  9-3;9-4(2-7)i]  =  23. 


MULTIPLICATION  71 

19.  Jamaica  one  year  exported  16,000,000  bunches  of 
bananas.  The  number  of  bunches  sent  to  the  United  States, 
less  600,000,  was  10  times  the  number  sent  to  all  other  coun- 
tries..  How  many  bunches  were  exported  to  the  United  States? 

20.  An  electric  light  company  expended  60^  for  every  dollar 
of  income.  Fuel  cost  5^  less  than  ^  as  much  as  other  things. 
What  was  the  cost  of  fuel  per  dollar  of  income  ? 

21.  Police  protection  in  a  large  city,  one  year,  cost 
$6,500,000  less  than  education.  The  total  expenditure  of 
the  city,  $98,100,000,  was  $6,600,000  more  than  3  times 
these  two  items.  What  sum  was  devoted  to  education  ?  to 
police  protection  ? 

22.  Cherries  brought  2^  more  per  pound  in  8-lb.  boxes  than 
in  5-1  b.  boxes,  and  a  5-lb.  box  sold  for  2^  less  than  ^  as  much 
as  an  8-lb.  box.     What  was  each  price  per  pound  ? 

23.  Upon  the  floor  of  a  room  4  feet  longer  than  it  is  wide 
is  laid  a  rug  whose  area  is  112  square  feet  less  than  the  area 
of  the  floor.  There  are  2  feet  of  bare  floor  on  each  side  of  the 
rug.     What  is  the  area  of  the  rug  ?  of  the  floor  ? 

24.  The  number  of  hundred  violets  sold  by  a  florist  during 
December  and  January  was  240.  The  price  per  hundred  was 
$2  in  December  and  $1|  in  January,  and  tlie  total  sum  re- 
ceived was  $  405.  How  many  hundred  violets  were  sold  each 
month  ? 

25.  A  party  of  8  traveled  second  class  from  London  to 
Paris  for  $  5.70  less  than  twice  the  amount  paid  by  a  party 
of  3  traveling  first  class.  If  a  first-class  ticket  cost  $4.15 
more  than  a  second-class  ticket,  find  the  price  of  each. 

26.  If,  in  coaling  the  British  battleship  Terrible  on  one 
occasion,  the  amount  loaded  per  hour  had  been  63  tons  less, 
the  \ime  taken  would  have  been  12^  hours.  If  the  amount 
per  hour  had  been  13  tons  less,  the  time  would  have  been  10 
hours.     How  many  tons  were  put  on  board  per  hour  ? 


DIVISION 


121.  In  multiplication  two  numbers  are  given  and  their 
product  is  to  be  found.  The  inverse  process,  finding  one  of 
two  numbers  when  their  product  and  the  other  number  are 
given,  is  called  division. 

10  -^  2  =    5,  and  2>  H-  c?  =  g 
are  inverses  of  5x2=10,  and  q  x  d=D. 

The  dividend  corresponds  to  the  product,  the  divisor  to  the 
multiplier,  and  the  quotient  to  the  multiplicand. 

Hence,  the  quotient  may  be  defined  as  that  number  which 
multiplied  by  the  divisor  produces  the  dividend. 

In  general,  the  quotient  of  any  two  numbers,  as  a  divided  by 

b,  indicated  hj  a-i-b,  or  -,  is  defined  by  the  relation 
b 

-  xb  =  a. 

0 

122.  Sign  of  the  quotient. 

The  following  are  direct  consequences  of  the  law  of  signs 
for  multiplication  (§  84)  and  the  definition  of  quotient : 

(+a)  (+^>)  =+a6;  .-.  +ab~{+b)= -\-a. 

(— a)  (-{- b)  =  -  ab  ;  r.  -  ab  ^  (-\-b)=  —a. 

(+  ^)  (—b)=  —  ab;  .:  —ab -h  {-b)  =  +a. 

(— a)  {— b)  = -\- ab ',  .-.  +  ab~r  (—b)  = —a. 

123.  Law  of  Signs  for  Division. — The  sign  of  the  quotient  ts  -f- 
when  the  dividend  and  divisor  have  like  signs,  and  —  ivhen  they 
have  unlike  signs. 

72 


DIVISION  73 

EXERCISES 

124.  1.   Divide  each  of  the  following  numl)ers  by  2. 

6,  -6,  10,  -10,  14,  -12,  -18,  22,  -8. 

2.  Divide  each  of  the  foregoing  numbers  by  —  2. 
Perform  the  indicated  divisions: 

3.  7)- 14.         4.   -3)li>.  5.   -3)- 12.         6.   -1)9. 
7.  4-5-(-4).     8.  22--(-2).     9.   -l-i-(-l).    10.   -(i-3. 

11.  56.  12.   ^.  13.    =4?.  14.  ^. 

4  —  i  6  —5 

125.  To  divide  a  monomial  by  a  monomial. 
Since  7  a  x  3  a^  =  21  tt\ 

by  def.  of  quotient,         21  a*  ^-  3  a*  =  7  a. 

The  quotient  may  be  obtained,  as  in  arithmetic,  by  removing 
equal  factors  from  dividend  and  divisor,  thus: 

o  a*       o 

126.  Law  of  Coefficients  for  Division.  —  Tlie  coefficient  of  the 
'/uotient  is  equal  to  the  coefficient  of  the  dividend  divided  by  the 
''^efficient  of  the  divisor. 

127.  Law  of  Exponents,  or  Index  Law,  for  Division. —  The  ex- 
ponent of  a  number  in  the  quotient  is  equal  to  its  exponent  in  the 
dividend  minus  its  exponent  in  the  divisor. 

Since  a  number  divided  by  itself  equals  1,  a^ -=- a*  =  a^-^  =  a'>  =  1 ; 
that  is,  a  number  whose  exponent  is  0  is  equal  to  1.    (Discussed  in  §  306.) 

The  law  of  exponents  for  division  is  of  general,  application, 
but  for  present  purposes  exponents  will  be  limited  to  posi- 
tive integers.     The  proof  for  positive  integral  exponents  follows ; 


74  DIVISION 

Let  m  and  n  be  positive  integers,  m  being  greater  than  n ;  and  let 
a  be  any  number. 

By  notation,  §  27,  a"^  =  a  ■  a  •  a  ••  to  m  factors, 

and  a^  =  a  ■  a  '  a  ■••  to  n  factors; 

g"*  _  a  '  a  ■  a  "■  to  m  factors 
a"      a  •  a  •  a  "'  to  n  factors 
Remove  equal  factors  from  dividend  and  divisor.     Then, 
a"*  ^  a"  =  a  •  a  '  a  ■-'  to  (m  —  n)  factors 
by  notation,  =  a"*-". 


EXERCISES 

128.   1.  5)5^ 

2. 

7cH')-S5c'd' 

3.    -4a3)-a^ 

5' 

-   bc'd 

l«^ 

Divide  as  indicated 

I: 

4.    2-}2^. 

5. 

3^-3*. 

6. 

a*)ai«. 

7.    22)2*. 

8. 

45^40 

9. 

af)af*. 

10.    28 -^^^c. 
—  4a6c 

11. 

-16ary^ 
4  ic?/% 

12. 

2  7rr 

4a26y 

14. 

-36ay;2« 
-9aV 

15. 

3  a6(a  -f  bf 
-2{a  +  b) 

4  a*6V 
*    20  a^dc^ 

17. 

-4xy;2* 

32  xy^^ 

18. 

2aKx-yY 
-<x-yy 

129.  To  divide  a  polynomial  by  a  monomial. 

Since,  §  93,  (a  +  b)x  =  ax-  +  bx, 

if  ax+fta;  is  regarded  as  the  dividend  (§  121)  and  x  as  the  divisor, 

(ax  -j-  bx)  -H  ic  =  a  +  6 ;  that  is, 

130.  The  quotient  of  a  polynomial  by  a  monomial  is  equal  to 
the  algebraic,  sum  of  the  partial  quotients  obtained  by  dividing 
each  term  of  the  polynomial  by  the  monomial. 

This  is  the  distributive  law  for  division. 


I 


DIVISION  76 

BXBRCISBS 

131.    1.    Divide  4  a^b  -  G  crfr  +  4  o^^  by  2  a6 ;  by  -  2  ab. 

PROCESS  PROCESS 

2  a5)4a«6-6a^6'H-4a6»        -  2  ab)     4  a«6  -  6  g'^  +  4  a6« 
2a»  ^Sab    -^'Jb'  -2a*  +3a6   -26* 

Test  of  Sigks.  —  When  the  divisor  is  positive,  the  signs  of  the  quotient 
sliould  be  like  those  of  the  dividend.  When  the  divisor  is  negative,  the 
4I1S  of  the  quotient  should  be  unlike  those  of  the  dividend. 

Test  of  Exponents.  —  Since  the  sum  of  the  exponents  in  each  term  of 
the  dividend  is  4,  and  the  sum  in  each  term  of  the  divisor  is  2,  the  sum 
(•f  the  exponents  in  each  term  of  tlie  quotient  should  be  4  —  2,  or  2.  ' 

Find  the  quotient : 

„     A  a^b^- 12  a'b'-^  16  a*b  ^     4m3?i- 8mV  +  4mw» 

4  a'6  4  mn 

24a«6'  +  32a^6^-40o^6*  5  a^  -  10  a^y' -\- 20  a^y' 

Sa*b'  *  *  5xy  ' 

^      -  35  a:*.vV  +  45  a^//«z*  g      _o-6-c-d-e 


5. 


5  srfz  - 1 

39  a^y^zf^  +  65  x^y'z'  -g^g^b-  ah  -  a*d  +  <r'p 

-ISxYzf'  '  '  _a 

10.  (34  a*aV  -  51  aVy*- 68  aV/)-^  17  ttV/. 

11.  (8  a^63  -  28  a*b*  - 16  a'b'  +  4  a*6«)  -^  4  a*b\ 

12.  [a(b  -  cy  -  b{b  -  cf  +  c(6  -  c)]  ^  (6  -  c). 

13.  [(x  -  y)  -  3(a;  -  y)*  +  4  a;(a;  -  yf]  h-  (a:  -  y). 

14.  («-H-2a?-">-5af+*-af+«  +  3«-+^)-*-«*. 

15.  (;/«-^i  _  2  ?/"-^*  -I-  y"+»  —  3  y*"^^  4-  y"+*)  -J-  y*-^\ 

16.  (a:"-a:»-»-har"---a^-«H-af-*-af-«)-s-a;*. 

17.  (r*"".^"  —  3  /*".s«"  —  5  7^«w>«)  H-  (_  5  r^s«"). 

18.  (0^-^*6*+*  —  a**+*6*-^  4-  a*'-'*6*+*)  -i-  a*'6'+*. 


76  DIVISION 

132.    To  divide  a  polynomial  by  a  polynomial. 


1.   Divide  3  0.-2- 

EXERCISES 

|-35  +  22a;by  a;-h5. 

PROCESS 

3x2_^22a;+35 
3  a;2  _^  15  a; 

a;  +  5 

TEST 

60-6 

X  times  {x  +  5) 

3a;  +  7 

=  10 

7  times  {x  +  5) 

7a;  +  35 
7  a; +35 

Explanation.  — For  convenience,  the  divisor  is  written  at  the  right  of 
the  dividend,  and  both  are  arranged  according  to  the  descending  powers 
of  X. 

Since  the  dividend  is  the  product  of  the  quotient  and  divisor,  it  is  the 
algebraic  sum  of  all  the  products  formed  by  multiplying  each  term  of  the 
quotient  by  each  term  of  the  divisor.  Therefore,  the  term  of  highest 
degree  in  the  dividend  is  the  product  of  the  terms  of  highest  degree  in 
the  quotient  and  divisor.  Hence,  if  3  x^^  the  first  term  of  the  dividend  as 
arranged,  is  divided  by  x,  the  first  term  of  the  divisor,  the  result,  3  x,  is 
the  term  of  highest  degree,  or  the  first  term,  of  the  quotient. 

Subtracting  3  x  times  (x  4-  5)  from  the  dividend,  leaves  a  remainder  of 
7  X  +  35. 

Since  the  dividend  is  the  algebraic  sum  of  the  products  of  each  term  of 
the  quotient  multiplied  by  the  divisor,  and  since  the  product  of  the  first 
term  of  the  quotient  multiplied  by  the  divisor  has  been  canceled  from  the 
dividend,  the  remainder,  or  new  dividend,  is  the  product  of  the  rest  of  the 
quotient  multiplied  by  the  divisor. 

Proceeding,  then,  as  before  we  find,  7  x  -f-  x  =  7,  the  next  term  of  the 
quotient.  7  times  (x  4-  5)  equals  7  x  +  36.  Subtracting,  we  have  no 
remainder.  Hence,  all  of  the  terms  of  the  quotient  have  been  obtained, 
and  the  quotient  is  3  x  +  7. 

Test.  — Let  x  =  1. 

Dividend  =  3  x2  +  22  x  +  35  =  3  +  22  +  35  =  60. 

Divisor =  x  +  5 =1+5 =6. 

Quotient  should  be  equal  to  10 

Quotient  =3x+7  =3  +  7  =10. 

Similarly,  the  result  may  be  tested  by  substituting  any  other  value  for  x, 
except  such  a  value  as  gives  for  the  result  0  ^  0,  or  any  number  divided 
by  0,  for  reasons  that  will  be  shown  in  §  547. 


DIVISION  77 

Rule.  —  Airmige  both  divideiid  and  divisor  according  to  the 
isceiiding  or  the  descending  powers  of  a  common  letter. 

Divide  the  first  term  of  the  dividend  by  the  first  tei-m  of  the 
lUvisor^  and  write  the  result  for  the  first  term  of  the  quotient. 

Multiply  the  whole  divisor  by  this  term  of  the  quotient,  and  sub- 
tract the  product  from  the  dividend.  Tlie  remainder  will  be  a 
new  dividend. 

Divide  the  new  dividend  as  before,  and  continue  to  divide  in 
this  way  until  the  first  term  of  the  divisor  is  not  contained  in  the 
fii'st  term  of  the  new  dividend. 

If  there  is  a  remainder  after  the  last  division,  write  it  over  the 
divisor  in  the  form  of  a  fraction,  and  add  the  fraction  to  the  part 
of  the  quotient  previously  obtained. 

Divide,  and  test  each  result : 

2.  ar^  4-  a;  -  20  by  a;  -f  5.         5.    rrr  -  18  -  3  m  by  m  -  6. 

3.  .r -h  7  a; -h  12  by  a; -h  3.       6.   a-- -f  15  a; +  54  by  a; -f  6. 

4.  l'-6P-lGhy  l^-\-2.       7.    10-lla;-f  ar^by  aj-lO. 
8.    81 +9a2-j-a^bva2_3a  +  9. 


PROCESS 

T?:ST 

a*  4-  9  a-  +  81 

a2_3a-|-9 

91-5-7 

a*-Sa'-{-9a^ 

a*  +  3  a  -f  9 

=  13 

3a«4-81 

3a«-9a«-h27a 
9a'-27a-| 
9a*~27a-f 

-81 
-81 

9.  a^-l-16-f  4a2by  2a-|-a2  +  4. 

10.  ar"'-61a;-60  by  aj2-2a;-3. 

11.  a*-41a-120by  a*4-4a-f-5. 

12.  2o3^-3(^-Sx-2x'hy  5a^-ix. 

13.  o«-}-a«-|-a*-f  a'^-|-3a-l  by  a-hl. 

14.  4y-9/-l-hGi/by3i/  +  2/-l. 


78 


DIVISION 


15.   2a'-5a^b-\-6a'b'-4.ab'-tb*  by  a'-ab  +  b\ 

PROCESS 


a^  -    ab  +  b^ 


2a^-Sab-^  b^ 


TEST 

0-1 
=  0 


2a'-5a^b-\-6  a'U'  -  4  aft^  +  b' 
2a*-2a^b+2a'b' 

-3a^b-\-Aa:'0'-4:ab^ 
-3a^b-\-3a'b^-3ab^ 

a%'-    ab^  +  b* 
a^-    ab^  +  b'' 

Note.  —  It  will  be  observed  from  the  test  that  0-^1=0.     In  general, 
0  -f-  a  =  0  ;  that  is,  zero  divided  by  any  number  equals  zero  (§  542). 

16.  6a2  +  13a6  +  662by  3a  +  26. 

17.  3  m^  —  4  amf  +  aV  by  am  —  1. 

18.  aa^  —  orx^  —  bx-  +  6-  by  ax—b. 

19.  20 x^y -25  a^ -IS y^ -\-27  xjf  hy  6y-5x. 

20.  a'^  —  Aa'x  +  6  a^x^  —  4  ax^  +  x*  by  a^  —  2  ax-[-  x^. 

a-1 


21.    a^  +  1 


o?-\-l 


if  +  a-  H-  a  + 1  4- 


a^  +  l 
a?—  a 


a-1 


22.   x^- 3  3^4-     cc-  +  2a;-l 


a;4 


a;« 


-  2  ar'  -h  3  ar^  +  2  a.- 

-  2  a:«  +  2  -T^  4-  4  a; 

aj2  _  2  a^  _  1 

a:^-     a;-2 
—    x  +  1 


a^- 

-      X-2 

a^- 

;1-  - 

-2.7^4-1  + 

-a^+1 
-x-2 

DIVISION  79 

Divide : 

23.  a^-h81  by  a;-3.  26.  a' -f  6' by  a -f  6. 

24.  r^-\-S2hy  x-\-2.  27.  m*  —  «*  by  m -f  w. 

25.  X*  —  y*  by  a^  4-  y*.  28.  7/i*  +  n*  by  m  +  n. 

29.  a^-foo*  — a*  +  2a  +  3bya  — 1. 

30.  x'-\-2o^-2  3i^-{-2x^-lhy  x-\-l. 

31.  2a^--a^  +  2aJ*-aj*  +  a^4-5  by  a;  +  l. 

32.  y*4-3y-h5/-f3/4-3?/4-5by  2/4-1. 

33.  2n*-4n*-3)i^-|-77i=^-3n4-2by  w-2. 

34.  1  by  1  +  a;  to  five  terms  of  the  quotient. 
36.  1  by  1  —  X  to  five  terms  of  the  quotient. 

36.  a»-6a«  +  12a-8-6«by  a-2-6. 

37.  /4-32ar*by  16a^  +  /-2a;^-8a:3^  +  4ar'/. 

38.  ar*  +  y -I- 2*  —  3  xyz  by  a;  4- y  4- 2. 

39.  m'' 4- n' 4- ar*  4- 3  m*n -f  3  mn*  by  m  4- n  +  a;. 

40.  a*  -  2  a^c  +  4  ac^—  aar*  -  4  c*a;  +  2  car'  by  a  -  a;. 

41.  a^  —  b^  -{-  c^  -\-S  abc  hy  a^  -]-  b'^  -i-  (^  -\-  ab  -  ac  +  be. 

42.  ^j^m*-\-im-inv^-^^-lm^hy  ^m^-m- f . 

43.  ^aV-f  aa^4-ta^-|a*by  Jar^  +  ^a^-^oa;. 

44.  |ar»4-^y*4-2^-ia;.yzby  ia;  +  J^2/  +  2. 
46.  r*»4-llr"4-30  by  r»4-6. 

46.  x^-^  -f  //*•+«  by  X"-*  4-  y"+^ 

47.  x*  4-  y"  by  X  4-  y  to  five  terms  of  the  quotient. 

48.  2-3  7i«4-13«*'4-23n»'-lln*'H-6w«'by24-3n'. 

49.  a*^*  4-  «'"^*  4-  a""*  by  0"+*  4-  a'  4-  a^*. 

60.  x*'+'/'H-2  x*'+V^»+x^+«y2.+2  by  a^y-i+af+2^. 

51.   6a*"  +  6a*— '-10 a**-* 4-20 a*" -«- 16 a*-*-^  by  2a'^-\- 
3  «"•-»  —  4  a—*. 


80 


DIVISION 


Divide,  using  detached  coefficients : 
52.   af  —  5x-\-4:hyaf  —  2x-j-l. 


PROCESS 


l  +  O-hO  +  0-5  +  4 
1-24-1 


2- 

-1  +  0 

2- 

-4  +  2 

3-2- 

-5 

3-6  +  3 

4- 

-8  +  4 

4- 

-8  +  4 

1-2  +  1 


1+2+3+4 


=  ar^  +  2a!2  +  3a:  +  4 


53.  x^  +  Sx  +  T  by  x--{-2x-^l.  56.   z^  -  64:  by  ;3-2. 

54.  a^  +  38a  +  12  by  a  +  2.  57.    ?i^  +  243  by  7i  +  3. 

55.  m^  — 19771- 6  by  m  +  2.  58.    a"*  — 256  by  a +  4. 

59.  a^-\-27a^-9a-10  by  o-Sa  +  al 

60.  21a;^  +  4-8a^-  +  6a;-29ar^  by  3a;-2. 

61.  16a^-lla^  +  2a'4  +  9-12a;  by  2a;-3. 

62.  30a;^-36a;  +  60a;2-62a^  +  8  by  5x-2. 

63.  x'^-2x^-x^-10x-36  by  x-2. 

64.  2/*  +  72/-10?/-2/'  +  15  by  y--2y-S. 

65.  7a^  +  2a^^-27a.-2  +  16-8a;  by  a;-  +  5a;-4. 

66.  28a;^  +  6a^  +  6a;2-6a;-2  by  2  +  2a;  +  4a^. 

67.  25if-20v^-^3v^-\-Uv-6  by  3^2-8^  +  2. 

68.  4.-18x-\-30x'-23a^-\-6x'  by  2x'-5x-}-2, 

69.  32a^  +  24a;^-25ic-4-16a;2  by  6x2-a;-4. 

70.  f-2t'-}-j\f-^if-\-j\t  +  i  by  ^-f. 

71.  ci^-Aa^  +  ija^-fi-a^  +  la-i  by  a-J. 

72.  ,6-^^+5^4_^3^.^37,2_^j^_^^^  by  ^^-1^  +  1 


DIVISION 


81 


SPECIAL  CASES  IN  DIVISION 
133.   By  actual  division, 


x-y 
x-y 


x  +  y- 


=a!*  +  ajy  +  y*. 


^  =Ql?  +  Ql^-\-xf-\-f. 


x  —  y 
x  —  v 


«*+a5^  +  a^/4-a^4- y*. 


Observe  that  the  difference  of  the  same  powers  of  two  num- 
bers is  exactly  divisible  by  the  difference  of  the  numbers. 


x-hy 

x  +  y 

ai*-y* 
x  +  y 


=  x  —  y. 

=  x^  —  xy-\-ff    rem.,  —2f. 

=  X^-QCh)-\-xf-f. 


— -^=x'^-3^y-\-x^f-xif^-\-y\    rem.,  -2^. 

\  x  +  y 

Observe  that  the  difference  of  the  same  powers  of  two  num- 
bers is  exactly  divisible  by  the  sum  of  the  numbers  only  when 
the  powers  are  even. 


=  x-\-yy    rem.,  2y*. 

x-y 

8.    -l^±^  =  «*4-a!y  +  2/',    rem.,  2/. 
x^y 

?^±i^'  =  x»  +  x*yH-a^  +  /,    rem.,  21^. 
x-y 

Observe  that  the  sum  of  the  same  powers  of  two  numbers  is 
not  exactly  divisible  by  the  difference  of  the  numbers. 
milne'8  stand,  alo.  —  6 


82 


DIVISION 


4. 


\t±f 


x-{-y 


x  —  y,     Tem.,2y\ 


=  oc^  —  xy-\-y\ 


x  +  y 

^  =a^  —  x^y  -i-  xy^  —  y^,    rem. ,  2  y*. 
x-\-y 

— ^tZ_  =  a?''  —  a?y  -\-  y?y^  —  x'lf  +  y^. 
x-\-y 


Observe  that  the  sum  of  the  same  powers  of  two  numbers 
is  exactly  divisible  by  the  sum  of  the  numbers  only  when  the 
powers  are  odd. 

134.  Hence,  when  ti  is  a  positive  integer, 

Principles.  —  1.   x^  —  y^  is  always  divisible  by  x  —  y. 

2.  a***  —  2/**  is  divisible  by  x  +  y  only  when  n  is  even, 

3.  x"^  -\-y'^  is  never  divisible  by  x  —  y. 

4.  a?"  +  ?/**  is  divisible  by  x-\-y  only  ivhen  n  is  odd. 
"Divisible"  means  "exactly  divisible." 

135.  The  following  law  of  signs  may  be  inferred  readily: 
Wlien  x  —  y  is  the  divisor,  the  signs  in  the  quotient  are  plus. 
When  x-\-y  is  the  divisor,  the  signs  in  the  quotient  are  alter- 
nately plus  and  minus. 

136.  The  following  law  of  exponents  also  may  be  inferred : 
The  quotient  is  homogeneous,  the  exponent  of  x  decreasing  and 

that  of  y  increasing  by  1  in  each  successive  term. 


EXERCISES 

137.   Find  quotients  by  inspection : 


2. 


a^- 

-b^ 

a  - 

-b 

m^  - 

-n^ 

4. 


m  —  n 


a-2 

a^  +  b^ 
a-\-b 


6. 


m^  +  n^ 
m-f-w 

c«4-27 
c  +  3 


DIVISION  88 


Find  quotients  by  inspection : 


a  — 5 


r—  8 


1+a 

a^-32 

x-2 

a'  + 128 

8.  t±^.  n.  5lnl.  14. 

71  +  4  n  —  1 

9.  25!±i''.  12.    ^.  16. 
m-f/i                            a;-|-l  a-\-2 

16.  Find  five  exact  binomial  divisors  of  a*  —  a^. 

Solution 

a«  —  x«  is  divisible  by  a  —  x  (l*rin.  1). 

cfi  —  2fi  is  divisible  by  a  -h  x  (Prin.  2). 

Since  (fi  —  yfi  =  (a^)*  —  (x-^)',  ««  —  a:«  may  be  regarded  as  the  differ- 
ence of  two  cubes,  and  is,  therefore,  divisible  by  a^  —  x'-^  (Prin.  1). 

Since  cfi  —  ofi  ■=  (a^)-*  —  (x*)"^,  a^  —  ofi  may  be  regarded  as  the  difference 
of  two  squares,  and  is,  therefore,  divisible  by  a*  —  3^  (Prin.  1). 

Since  cfi  —  x"^  =  (a^)^  —  (x^)%  cfi  —  ofi  may  be  regarded  as  the  differ- 
ence of  two  squares,  and  is,  therefore,  divisible  by  a*  +  x^  (Prin.  2). 

Therefore,  the  exact  binomial  divisoi-s  of  a"^  —  x^  are  a  —  x,  a  +  x, 
a^  —  x^,  a'  —  x*,  and  a'  +  x*. 

17.  Find  an  exact  binomial  divisor  of  a*  4-  a^. 

SOLUTIOV 

Since  «•  +  x«  =  (a^)*  +  (x*)',  a*  +  x*  may  be  regarded  as  the  sum  of 
the  cubes  of  a^  and  x^,  and  is,  therefore,  divisible  by  a*  +  x*  (Prin.  4). 

Find  exact  binomial  divisors : 

18.  a^-m^  24.  a^  +  a\  30.  a^-6Sfour. 

19.  a'^-m*.  25.  a'^-hft*".  31.  a«-l,  five. 

20.  6»-|-a5».  26.  a^^  +  ft*.  32.  a^-fc",  six. 

21.  a^-a\  27.  a"  4-6".  33.  a^«-6'^five. 

22.  c»4-7l^  28.  o^-27.  34.  a^*-^^*,  eight. 

23.  a«  +  6*.  29.  a«-27.  35.  a^--6^2  nine. 


84  DIVISION 


Algebraic  Representation 

138.  1.  Express  m  dollars  in  terms  of  cents ;  m  cents  in 
terms  of  dollars. 

2.  Find  the  value  of  x  that  will  make  6  x  equal  to  48. 

3.  By  what  number  must  25  be  multiplied  to  produce  300  ? 
10  to  produce  a:  ?  r  to  produce  s  ? 

4.  Eepresent  (the  third  power  of  a  minus  the  fifth  power 
of  X)  divided  by  (m  plus  ?i^). 

5.  Express  the  multiplicand  when  Imn  is  the  product  and 
Im  the  multiplier. 

6.  Find  an  expression  for  5  per  cent  of  ic ;  i/  per  cent  of  z. 

7.  It  takes  a  men  c  days  to  do  a  piece  of  work.  How  long 
will  it  take  one  man  to  do  it  ?  2  men  ?  x  men  ? 

8.  At  a  factory  where  iV  persons  were  employed,  the  weekly 
pay  roll  was  P  dollars.  Find  the  average  earnings  of  each 
person  per  week. 

9.  A  train  ran  i)[f  miles  in  ^  hours  and  m  miles  in  the  suc- 
ceeding h  hours.  Find  its  average  rate  per  hour  during  each 
period  and  during  the  whole  time. 

10.  A  farmer  has  hay  enough  to  last  m  cows  for  n  days. 
How  long  will  it  last  {a  —  5)  cows  ? 

11.  Indicate  the  quotient  of  m  +  ^  divided  by  the  number 
whose  first  digit  is  x^  second  digit  y^  and  third  digit  z. 

12.  A  dealer  bought  n  50-gallon  barrels  of  paint  at  c  cents 
per  gallon.  He  sold  the  paint  and  gained  g  dollars.  Find  the 
selling  price  per  gallon. 

13.  If  it  takes  6  men  c  days  to  dig  part  of  a  well,  and  d  men 
e  days  to  finish  it,  how  long  will  it  take  one  man  to  dig  the 
well  alone  ? 


DIVISION  85 

Equations  and  Problems 

139.    1.    Find  the  value  of  x  in  the  equation  bx  —  b*  =  cx  —  c*. 

Solution 
bx  -  b^  z=  ex  -  (^. 

Transposing,  bx  -  <x  =  h'^  -  c*. 

Collecting  coefficients  of  a:,      (b  —  c)x  =  6*  —  c-*. 

Dividing  by  6  -  c,  x  =  =  6  +  c. 

b  — c 

2.   Find  the  value  of  x  in  the  equation  x  —  a^  =  2  —ax. 

Solution 

a;  —  a*  =  2  —  aa;. 

Transposing,  aa;  +  a:  =  a»  +  2. 

Collecting  coefficients  of  x,      (a  +  l)a:  =  a^  H-  2. 

Dividing  by  a  +  1,  x  =i  ^^^t.?  =  a«  -  a  +  1  +  —^  • 

a+1  a+ 1 

Find  the  value  of  a;  in  : 

3.  cx-c»-crH-(ia;  =  0.  6.  7  a-lO^a'-ax-hBx. 

4.  a2-aa;-2a6  +  6a;-f  6'^  =  0.  7.  x -l-c  =  cx-c^ -c\ 
6.  2n*-|-5n  +  x=7i*  — wic  — 2.  8.  2m''  — wa;-f  nx  — 2w*=0. 

9.  :iab-a^-2bx  =  2b^-ax, 

10.  a«aj-a»-h2a2-l-5a;-5a-hl0  =  0. 

11.  ax-2bx-{-Scxz=ab-2b^-{-3bc. 

12.  cx-c^-2c*-2c*  =  2c-a;-hl. 

13.  9a2-|-4?MX= -(3aa;-16m*). 

14.  a;  -f  6  w*  —  4  71''  =  1  —  3  7ix  -f  2  n  -  n*. 
16.  n*x  —  3  wiV  4-  nx  -f  3  ?h2  -f  a;  =  0. 

16.   a;-36*-1926V-4cx  +  16c*a;=:0. 


86  DIVISION 

Solve  the  following  problems  and  verify  the  solutions : 

17.  George  and  Henry  together  had  46  cents.  If  George 
had  4  cents  more  than  half  as  many  as  Henry,  how  many 
cents  had  each? 

Solution 

Let  X  =  the  number  of  cents  George  had. 

Then,  x  —  4  =  the  number  of  cents  George  had  less  4, 

and  2(ic  —  4)  =  the  number  of  cents  Henry  had  ; 

.•.a;  +  2(a;-4)=46. 

Solving,  X  =  18,  the  number  of  cents  George  had, 

and  2(ic  —  4)  =  28,  the  number  of  cents  Henry  had. 

Verification 

The  answers  obtained  should  be  tested  by  the  conditions  of  the  prob- 
lem. If  they  satisfy  the  conditions  of  the  problem,  the  solution  is  pre- 
sumably correct. 

1st  condition  :  They  had  together  46  cents. 

18  -H  28  =  46. 

2d  condition  :  George  had  4  cents  more  than  half  as  many  as  Henry. 

18  =  I  of  28  +  4. 

18.  In  a  certain  election  at  which  8000  votes  were  polled,  B 
received  500  votes  more  than  half  as  many  as  A,  How  many 
votes  did  each  receive  ? 

19.  A  had  $40  more  than  B;  B  had  $10  more  than  one 
third  as  much  as  A.     How  much  money  had  each  ? 

20.  In  2  years  A  will  be  twice  as  old  as  he  was  2  years  ago. 
How  old  is  he  ? 

21.  Two  wheelmen  start  at  the  same  time  from  A  to  ride  to 
B.  One  rides  at  the  rate  of  10  miles  an  hour,  and  rests  3 
hours ;  the  other  rides  at  the  rate  of  8  miles  an  hour,  and  by 
resting  only  1  hour  arrives  at  B  as  soon  as  the  faster  rider. 
How  many  hours  are  occupied  in  making  the  trip  ?  How  far 
is  it  from  A*  to  B  ? 


DIVISION  87 

22.  It  cost  a  man  60^  to  send  a  telegram  at  "  30-2  ",  that  is, 
30^  for  the  first  10  words  and  2^  for  each  additional  word. 
How  many  words  did  the  message  contain  ? 

Solution 
I^t  X  be  the  number  of  words  in  the  message. 
Then,  x  —  10  will  represent  the  number  of  words  in  excess  of  10  words. 

.•.30  +  2(x-  10)=  60. 
Solvibg,  X  =  25,  the  number  of  words. 

Verification 
30^  for  10  words  +  30^  for  15  additional  words  =  60^. 

23.  How  many  words  can  be  sent  by  telegraph  from  New 
Haven  to  New  York  for  75^  at  the  day  rate,  "  25-2  "  ? 

24.  A  long-distance  telephone  message  cost  me  $1.25.  The 
rate  was  50^  for  the  first  3  minutes  and  15^  for  each  additional 
minute.     How  long  did  the  conversation  last  ? 

25.  The  day  rate  for  a  telegram  between  New  Orleans  and 
New  York  is  "  00-4  "  and  the  night  rate  is  "  40-3."  A  mes- 
sage of  a  certain  number  of  words  cost  25^  less  to  send  at 
night  than  in  the  daytime.     Find  the  number  of  words. 

26.  Separate  24  into  two  parts,  x  and  24  —  x,  such  that  one 
part  shall  be  3  less  than  twice  the  other. 

27.  Separate  52  into  two  parts  such  that  2  times  one  part 
shall  be  4  greater  than  3  times  the  other. 

28.  Mary  bought  17  apples  for  61  cents.  For  a  certain 
iininl)er  of  them  she  paid  5  cents  each,  and  for  the  rest  she 
l)aid  3  cents  each.     How  many  of  each  kind  did  she  buy? 

29.  George  is  i  as  old  as  his  father ;  a  years  ago  he  was  ^ 
as  old  as  his  father.     What  is  the  age  of  each  ? 

30.  A  rug  3  feet  longer  than  it  is  wide,  placed  on  the  floor 
of  a  certain  room,  leaves  a  margin  of  2  feet  on  every  side.  If 
the  area  of  the  floor  is  172  square  feet  greater  than  the  area 
of  the  rug,  what  are  the  dimensions  of  the  floor  ? 


88  REVIEW 

REVIEW 

140.    1.    Define  a  term  ;  similar  terms ;  the  degree  of  a  term  ; 
the  degree  of  an  expression. 

Illustrate  symmetrical  expression  ;  homogeneous  expression. 

Simplify  : 

2.  37^+2 a-Vxy  —  Smn  —  4: x^—o aVa^2/+3 x^+iaVxy-\-4:mn. 

3.  5x^+3x^y-i-4:xy''—f--Vx-\-6y^-i-xy'-Vy-5x^y-7a^ 
-5xy'-]-a^-{-2Vx  +  x''y-6f-\-^y-Vx-2x'y-\-3xy'-h  a^. 

4.  How  may  a  parenthesis  preceded  by  a  minus  sign  be 
removed  from  an  algebraic  expression  without  changing  the 
value  of  the  expression  ? 

Simplify : 

5.  x^  —  (x'^  —  D  x^y  4- 10  o^f  -  10  x^f  -}- 5  xy^  —  f). 

6.  |a-|a.'-(}a-ia^)-(3  6--Va^-fa)  +  ia. 


7.  X' ~(2xy  —  y-)  —  {x^-\-xy  —  y'^)—xr  —  2xy  —  y- -\- 5 y"^. 

8.  7n-\-2\2m—[n  +  3p  —  (4|)  —  3  ?i)  -  5  n  +  2  m]  -  7^1 . 

9.  What  are  the  various  ways  of  indicating  multiplication 
in  algebra? 

Expand : 

10.  {7n  —  x){m  +  x).  14.  (a"* +  &")(«"  — ft"*). 

11.  (x-  +  4)(a;2-3).  15.  (a  +  6  +  c)(a  +  5-c). 

12.  (x^  +  x^){x-\-V).  16.  {x-\-y  +  z){x  —  y-{-z), 

13.  {x  —  V)(l-{-x).  17.  {m-\-n—p){m  —  n-\-p). 

18.  Why  should  the  terms  of  the  dividend  and  divisor 
usually  be  arranged,  before  division,  according  to  the  ascend- 
ing or  the  descending  powers  of  some  letter  ? 

19.  What  is  the  advantage  of  using  detached  coefficients  ? 


REVIEW  89 

Arrange  terms  and  divide,  using  detached  coefficients: 

20.  ar^-a;  +  2ar''-8-2ar*4-12ar*by  x-fl. 

21.  a;*-'4a;  +  5a^-4ar'+l  by  l-3x  +  a^. 

22.  tt"-12a2-a4-12by  a^-S-f  4a-2a-. 

Simplify : 

23.  l_5l_[a:2_3_(2a;-4)2  +  3r^  +  l]-(a;-4)2j-l. 


24.  x-\ox-[6x-(7x-Sx-9x)-10x2-^llx\-\-9x, 

25.  l_5_[_(l_x)_l]_lj_5a;-(5-3a;)-74-a;|. 

Collect,  in  order,  the  coefficients  of  ic,  i/,  and  z : 

26.  ax -\- ay -\- az  —  bx  —  by  —  bz. 

27.  ax-2y  +  cz  +  by -12  x-^iz. 

28.  3  ma;  —  nx  +  6y  —  y  +  3  C2  —  4  2. 

29.  py  —  y  —  4z-\-bz  —  x  +  mx  —  nx  —  z. 

30.  ex  —  by  —  S  az  -\-  x  —  y  —  4  z  -^  z  —  y. 

31.  State  the  law  of  signs  for  multiplication ;  for  division. 

32.  What  is  the  sign  of  the  product  of  an  even  number  of 
negative  factors  ?  of  an  odd  number  of  negative  factors  ? 

Expand: 

33.  {a-b){a-\-b)(a^  +  b*). 

34.  (l-x)(l-far)(l  +  a^(l+ar*). 
36.  (1  -  x)0  +  x)(l  -x)(l-\-  x). 

36.  ((^  +  Sah/-\-Saf)(a'-2ay-\-f). 

37.  (a*»H-2af2r  +  /')(aJ**-2x»y*  +  y'"). 

38.  (ia?  +  ixy  +  ^y')(\x^-^xy  +  ^f). 

39.  (.2  a»  -  .8  a  -f-  1.6)(.l  a«  +  .4  a  +  .8). 


90  REVIEW 

40.  Give  a  rule  for  multiplying  a  monomial  by  a  monomial ; 
for  dividing  a  polynomial  by  a  polynomial. 

41.  State  and  illustrate  two  ways  of  testing  the  correctness 
of  a  result  in  algebraic  multiplication  ;  in  algebraic  division. 

Expand,  using  detached  coefficients  ;  test  results : 

42.  (a^-f  a^  +  «2  +  «-f-l)(a  — 1). 

43.  (oi:^  —  x*-\-a^  —  x^-\-x  —  l)(x-^l). 

44.  (a^  +  2  a^  +  4  a^  +  8  a^  + 16  a  4-  32)  (a  -  2) . 
Divide,  and  test  results  : 

45.  4.-10b^-5b-\-b^hjSb-2b'  +  b''-l. 

46.  miO-6?7i3-u5m-2  by  2  m^ -2 -{-m'-3m. 

47.  127  a'^-20  a  +  a^-lOOa^+ie-ieOa^  by  a3-6a2+5a-4. 

48.  b''>-\-29b'-22-61b~+210b-170b^  by  b'-5b+2  b'-ll. 

Simplify : 

49.  a-(2  6  +  5a)(6  6-3a)-26-6[3a2-4a6-2&2]. 

50.  x-l3y-}-[4:X-2(j/-i-3x)-~3yy-(oy-i-2xy-Sy}. 

51.  (a' 4-  ab  -  by  -  (a'  -  ab  -  b'f  ~  4  ab(a'  -  b'). 


52.  a'-l-b''-\-b{5b-3d)-l-3ab-\-d'-b(a-2a-\-2b)\\ 
Square : 

53.  2x  —  3y.  56.    10  —  3  ic.  59.    a4-b  —  c-^d. 

54.  x^  —  aaf.  57.    ri'  —  Tny.  60.    2  a  — 3  6  — 4  c. 

55.  oa?  —  l.  58.    7  x —  b'^y.  61.   ic'*-^  -  2/ —  a^. 

62.  What  laws  are  illustrated  by  a(bc)  =  6(ac)  ? 

63.  Show  that  a^  •  a®  =  oP^ ;  that  a^^  —  a^  =  al 

64.  In  what  respect  do  (a  —  b)  and  (6  —  a)  differ  ?     Expand 
and  compare  (a  —  6)^  and  (5  —  a)^ 


REVIEW  91 

Collect,  in  order,  the  coefficients  of  a?,  y,  and  z : 

65.  lQny  —  \Qmx-\-ax-\-by-\-cx  —  2  y. 

66.  rnx -\'ny-\-az-\-2ax  — 2 my  +  2 nz. 

67.  x  —  y  —  az-{-^mx-^aby  —  3^-\-lf-\-z. 

68.  a*x  -\-l^y  —  2  ax  —  2  cz  +  (?z  •\-  X  -\-  y  -\-  z. 

69.  m*a;  —  n^y  +  m-y  —  7i^ic  —  2  W7ia  —  2  mny  —  n'^z  +  z. 

70.  4(ax  —  63/  +  cz)  —  2(6a;  —  ay—  dz)  —  2{x  —  y  +  z). 

71.  Show  why  a  broader  definition  is  necessary  for  multi- 
I  plication  in  algebra  than  in  arithmetic. 

Expand : 

72.  (5a-4y)(5a-32/).  75.  (2a^x-bhhi){^o:'x-Sb^y), 

73.  (6x  — 4iy)(3a:-|-5i/).  76.  (6amn-f5j9)(6awiw— 3/)). 

74.  (3a;  +  a3/)(3a;  +  6y).  77.  (3a-+»-2  6-»)(2a"+^-36-i). 

78.  {x-\-y)(x-y){^^y%7^-{-y'){7?  +  f), 

79.  (m*  +  l)(m*  +  l)(m*  -f  l)(m  +  l)(m  —  1). 

80.  (16«*4-l)(4a^  +  l)(2a;  +  l)(2a:-l). 

81.  For  what  values  of  n  is  af'-iry*'  divisible  by  a;-|-y?  by 
x  —  y?     When  is  a*  —  y"  divisible  bya;  +  y?  bya;  —  y? 

82.  State  the  law  of  signs  for  the  quotient  when  «*  +  y"  or 
a:"  —  y*  is  divided  by  x  +  y  ovx  —  y\  the  law  of  exponents. 

Divide : 

83.  a?'-'^  -  3f+Y  -  2  y*-  +  3  y-z*-*  -  z^"*  by  x-^^+y*  -  z-^ 

84.  6aJ  +  ^ay-iia^  +  i\y'hya'-\-ia'y--iaf-^lf. 

85.  a*c-a6*  +  acd  — ad*  — a^4-&'— 6cd+M*-ac*  +  c^  — c*<i 
-f  cd*  by  aq  —  6*  +  cd  —  d*. 


94  FACTORING 

2.  Factor  ax  —  ay  —  hx-\-  by. 

Solution 
ax  —  ay  —  bx  -^  by  =  a(x  -  y)—  b(x  -  y) 

=  (a-b)(x  -  y). 

Observe  that,  when  the  first  two  terms  are  factored,  (x  —  y)  is  found 
to  be  the  binomial  factor.  Since  {x  —  y)  is  to  be  a  factor  of  the  other  two 
terms,  the  monomial  factor  is  — &,  not  +6,  for  (  —  bx-\-by)-~(x—y)  =  —  b. 

3.  Ysictor  ex -{- y  —  dy -\- cy  —  dx -^  X. 

Solution 
ex  -\-  y  —  dy  -\-  cy  —  dx  -f  x 
Arranging  terms,  =  ex  —  dx  +  x  +  cy  —  dy  +  y 

=  (c-d  +  l)x+(e-d+  l)y 
=  (c-d-\-  l){x  +  y). 

Factor,  and  test  each  result,  especially  for  signs : 

4.  am  —  an -\- mx  —  nx.  18.  x^  +  x^  -\- x^y  -\-y. 

5.  hc  —  hd-\-cx—dx.  19.  2  — 2?i  — n^  +  n^ 

6.  pq  —  px  —  rq  +  rx.  20.  x^  —  x  —  a  +  ax. 

7.  ay  —  hy  —  ab  +  ¥.  21.  3ar^— Iooj  +  IOt/  — 2  a;V 

8.  x:^  —  xy  —  bx  +  by.  22.  12a^  -  8  c(6  —  3  a^  +  2  a-6. 

9.  W  —  bc-\-ah  —  ac.  23.  3  m^n  —  9  mn^  +  am  —  3  an. 

10.  X-  +  xy  —  ax  —  ay.  24.  15  ab-  —  9b^c  —  35  ab +  21  be. 

11.  C-  — 4c-hao  — 4  a.  25.  16ax-^12ay  —  Sbx  —  6by. 

12.  2x  —  y-{-4:X^  —  2xy.  26.  «x^  —  ax  — aa;?/  + a?/ -fa;  — 1. 

13.  1  —  m  +  n  —  m/i.  27.  xy  -4-  .r  —  3  /  —  3  ?/  —  4  ?/  —  4. 

14.  2p-\-q-\-6p^ -^Spq.  28.  ax  — a— 6x  + 5  —  ex  +  c. 

15.  ar  —  rs  —  a6 -f  &s-  29.  mx  —  nx  —  x  —  my-[-ny-\-y. 

16.  x^4-x2  +  x  +  l.  30.  bx^  —  b  —  xy  —  y+yx^—bx. 

17.  2/^  +  y- —  32/  — 3.  31.  m^  +  m/i  4- m/i  +  ?i^  +  ??i  +  71. 


FACTORING  96 

148.  To  factor  a  trinomial  that  is  a  perfect  square. 
Since  by  multiplication,  §§  105,  108, 

(a  +  6)(aH-6)=a'  +  2a6-f  62and(a-6)(a-6)=a'-2a6H-6«, 

cr  +  2ab  +  b^=(a  +  b)(a-^b)  and  a*  _  2 a6  +  b^=(a-b) (a-b). 

These  two  trinomials  are  perfect  squares,  for  each  may  be 
separated  into  two  equal  factors.  They  are  types,  showing 
the  form  of  all  trinomial  squares,  for  a  and  6  may  represent 
any  two  numbers. 

149.  A  trinomial  is  a  perfect  square,  therefore,  if  these  two 
conditions  are  fulfilled : 

1.  Two  terms,  as  4-  a^  and  -h  b^,  must  be  perfect  squares. 

2.  The  other  term  must  be  numerically  equal  to  twice  the 
product  of  the  square  roots  of  the  terms  that  are  squares. 

26x2  -  20xy  +  4y2  is  a  perfect  square,  for  26x2  =(6x)2,  4y2  =(2y)2, 
and  -  20  xy  =  -  2(6  x)<2  y) . 

150.  Every  number  has  two  square  roots,  one  positive  and 
the  other  negative.  In  factoring,  usually  only  the  positive 
square  root  is  taken. 

Thus,  \/26  =  5  or  -  5,  for  5  .  6  =  26  and  (-  6)(-  6)=  26. 
a2+  2a6  +  62  =(a  +  6)(a  +  h)  or  (— a  -  6)(- a  -  6),  but  we  usually 
factor  trinomial  squares  in  the  first  way  only. 

Rule.  —  Connect  the  square  roots  of  the  t£rms  that  are  squares 
with  the  sign  of  the  other  term,  and  indicate  that  the  result  is  to  be 
taken  twice  as  a  factor. 

From  any  expression  that  is  to  be  factored,  the  monomial 
factors  should  usually  first  be  removed. 

Thus,  2o8-4a2  +  2a  =  2a(a«-2a  +  l)  =  2a(a-l)2. 

EXERCISES 

151.  Factor,  and  test  each  result: 

1.  3i?-\-2xy-^y^.  4.   m^  —  2mn  +  n^. 

2.  p^-2pq-\-q'.  6.    a^-2x-\-l. 

Z.    c^-{-2cd-{-d^.  6.    a:2  +  4x  +  4. 


96  FACTORING 

7.  a^-|-6a;  +  9.  18.    W p^ - 24. j) -h 9. 

8.  4  -  4  a  4-  a^.  19.    9  x^  -  A2  x -\-  49. 

9.  4a-4a2  +  a^  20.   9  +  42  Z>3  +  49  56. 

10.  m--Sm-\-lQ.  21.  9m»-6m^  +  l. 

11.  a^- 16  a +  64.  22.  4.  xY  -  20  xy  +  25. 

12.  5  x-^  4- 30  a; +  45.  23.  4  a:2_j_i2a;?/2  +  9?/V^ 

13.  3a^  +  6xy-t-3f.  24.  9a'V-6am+l. 

14.  2m2-4mn  +  2n2.  25.  2 a;  +  20 a^o;  +  50 a^a;. 

15.  1  +  45  +  461  26.  18  a^ft  4. 60  aft^  +  50  R 

16.  l-6(x3  +  9al  27.  a^x^ -2a:x?hf -\-hhf. 

17.  10  x^  —  20  a;  + 10.  28.  a^'^^"  —  2  a;"?/"^;"  + 1/^''^;^". 

When  either  or  both  of  the  squares  are   squares  of  poly- 
nomials, the  expression  may  be  factored  in  a  similar  manner. 

29.  Factor  0^ ^Qx{x-y)-{- %x - y)\ 

Solution 

=  [«  +  3(x  -  y)^[_x  +  3(x  -  y)] 
=  (a;  +  3x-3i/)(a;  +  3x-3y) 
=  (4a;-  3?/)(4a;-3y). 

30.  Factor  (a  -  hf  +  2(a  -  5)(5  -  c)  +  (6  -  c)^. 

Solution 

(a-  6)2  +  2(«-&)(&-c)  +  (?>-c)2 

=  [(«  -  ft)  +  (&  -  c)][(«  -  &)  +  (^  -  c)] 
=  (a  -  6  +  &  -  c)(a  -h  +  h  -c) 
=  (a  —  c)(a  —  c). 

Test.  —  When  a  =  3,  ft  =  2,  and  c  =  1, 

(a  _  6)2  +  2(a  -  6)(6  -  c)  +  (&  -  c)2  ==  12  +  2  •  1  •  1  +  I2  =  4, 
and  (a  —  c)  (a  —  c)  =  2  •  2  =  4. 

Factor,  and  test  each  result : 

31.  x^-^2x{x  —  y)  +  (x  —  yy.    33.    (r  +  s)2  —  4(r  +  .s)  +  4. 

32.  t'-At{t-l)+4.(t-iy.     34.    c--6c(a-c)+9(a-c)2. 


FACTORING  97 

35.  16-24(^-0  4-9(^-0'. 

36.  U{x-'y)^(x-yy  +  49. 

37.  (a  +  by-2(a  +  b)(b  +  c)  +  (b-^cy. 

38.  (a  -  2  xy  +  4(a  -  2  x){2  x  -  6)  +  4(2  x  -  by. 

39.  16(a  -  af  +  32(a  -x)(xi-b)-\-  16{x  +  6)«. 

40.  (a  +  3  6)*  -  4(a  +  3  6)  (3  6  -  2  c)  -f  4 (3  6  -  2  c)«. 

41.  (a^4.aj  +  l)2  4.2(a;  +  l)(a^  +  x+l)-h(«-fl)^ 

42.  (a  -f  6  +  c)*  -f  2(a  +  ft  -  c)(a  +  6  -h  c)  -h  (a  +  i;  -  c)«. 

152.  To  factor  the  diiference  of  two  squares. 
By  multiplication,    (a  -\-b){a  —  b)  =  a^  —  b\ 
Therefore,  a^-b^=  {a-\-b){a~b). 

Rule.  —  Find  the  square  roots  of  the  two  tei'ms,  and  make 
their  sum  one  factor  and  their  difference  the  other. 

Sometimes  the  factors  of  a  number  may  themselves  be  factored. 

EXERCISES 

153.  1.  Factor  6^-/. 

Solution.  h'^  -  y"^  :=  (Jb  -\-  y) (Jb  —  y). 

2.  Factor  iB*~l. 

Solution.  x'^  -  \  =  {x -\-\){x-~\). 

3.  Factor  a^--l. 

Solution.  ac<  -  1  =  (a;2  +  1 )  (x2  -  1) 

=  (a:2  +  l)(3r  +  l)(a;-l). 

Resolve  into  their  simplest  factors : 

4.  :t?-m\  10.  a:^-81.  16.  25ic*-l. 

5.  a^-f.  11.  a'-b\  17.  144m*-l. 

6.  a^-ie.  12.  ai«-6».  18.  36  a* -225. 

7.  a2-9.  13.  9a2-49  6*.  19.  12162-aV. 

8.  25 -c^.  14.  aV-4c*.  20.  100  a*- 81  y». 

9.  a^-49.  15.  m*-Un\  21.  64  ar^  -  625  .y^. 

MILNK*S    STAND.    ALO. 7 


98  FACTORING 

22.  169  -  a-c\  27.  4  m^  -  4  b\  32.  x"  -  .01. 

23.  400a;2_35^2  28.  3x'-3y^  33.  a*-yV 

24.  144m2-16n2.  29.  5a^-5.  34.  ^-f\ 

25.  rcy  — 256.  30.  3a^  — 3  a.  35.  x-^"-^  — 3/^. 

26.  2a^  — 2  61  31.  oj^  — a^^  36.  ar'"+i  — a;/". 

When  either  or  both  of  the  squares  are  squares  of  poly- 
nomials, the  expression  may  be  factored  in  a  similar  manner. 

37.   Factor  25  a^  _  (3  a  +  2  6)1 

Solution 
One  factor  is  5  «  +  (3  «  +  2  6)  and  the  other  is  5  a  —  (3  a  +  2  &). 
6a+C3a  +  2  6)  =  5a  +  3a  +  26  =  8a  +  26  =  2(4a  +  6). 
6a-(3a  +  2  6)  =  5«-3a-26  =  2a-26  =  2(a-6). 
.-.  25  a2  -  (3  a  +  2  6)2  =  2(4  a  +  &)2(a  -  6) 
=  4(4a  +  fe)(a-6). 
Factor : 

38.  a^-ip-^-cf.  42.    9  6^  _  («  _  3^)2. 

39.  52 -(2  a +  6)2.  43.    9a2-(2a-5)2. 

40.  a2-(a  +  6)2.  44.    a:4-(3x2-2?/)2. 

41.  4c2-(6  +  c)2.  45.   49a2-(5a-4  6)2. 

46.  Factor  (3  a- 2  6)2- (2  a-5  6)2. 

Solution 

(3a- 2  &)2- (2  a -56)2 
=  [(3a-2  6)  +  (2a-56)][(3a-2  6)-(2a-5  6)] 
=  (3a-2&+2a-5  6)(3a-26-2a  +  5  6) 
=  (5a-76)(a  +  36). 

Factor : 

47.  (2a  +  36)2-(a  +  6)2.  49.    (2  ic -f  5)^  -  (5  -  3  a;)2. 

48.  (5  a -3  6)2 -(a- 6)2.  50.    (a  -  2  6)2  -  (a  -  5)2. 


FACTORING  99 

61.    (2x-3yy-(Sy-\-zy.  54.    (9x  +  6yy  -  (4x-3yy 

52.  (56 -4c)^- (8a -2c)-.        55.    (x" -\- x'y -  (2 x  +  2y 

53.  {Ax-Syy-(2x-3ay.       56.    (a  +  6 +  c/-(a -6-c)*. 

67.  Factor  a*  +  4  —  c*  —  4  a. 

Solution 
a*  +  4  -  c2  -  4  a 
Arranging  terms,  =  (a-^  —  4  a  +  4)  —  c* 

=  (a  -  2)2  -  c2 
=  (a  -2  +  c)(a  -  2  -  c). 

68.  Factor  a2  4-62-c*-4-2a6-i-4c. 

Solution 
a-i  +  62  _  c2  -  4  -  2  aft  +  4  c 
Arranging  terms,  =  a*  —  2  a6  +  6*  —  c"^  +  4  c  —  4 

=  (a*  -  2a6  +  ft^  _  (c*  -  4  c  +  4) 

=  (a  _  />)-2  -  (c  -  2)2 

=  (a  -  6  +  c  -  2)(a  -  6  -  c  +  2). 

Factor,  and  test  each  result : 

69.  a*-2ax-\-a^-n\  65.    b- ~3r -  f-h2  xy. 

60.  6=^4-2  6j/+2r-    «^.  66.  ic^  -  x- -y- -2xy. 

61.  1  _4y -r4^'-a2.  67.  9  c'-ar^-y^-f  2a^. 

62.  r-2rx-\-a^-16t'-.  68.  ar' -  a^a;  -  4  fe^x  -  4  a6a;. 

63.  9a^b-6a^-\-b^-ihcr.  69.  6c=^  -  9  a^fe  -  6«  -  6  aft^. 

64.  (r-a'-62-2a6.  70.  ab' -  4 a^- 12 a^c- 9 ac^. 

71.  a2-2a6  +  //-c2-f2cd-(f*. 

72.  x'-2xy-{-f-  m*  -}- 10  m  -  25. 

73.  4x^-{-9~12x  +  10mn-m^-25n\ 

74.  x^  —  a-  +  y^  —  6-  4-  2  x^  —  2  a6. 


100  FACTORING 

154.  To  factor  a  trinomial  of  the  form  jr-  +  yojr  -f-  q. 
By  multiplication, 

(x  +  a){x  ■^h)  =  x^  +  {a  +  h)x-\-db. 

This  trinomial  consists  of  o^,  an  a>term^  and  an  absolute 
term ;  and  therefore  has  the  type  form  xr  +px  +  q. 

Therefore,  by  reversing  the  process  of  multiplication,  a  tri- 
nomial of  this  form  may  be  factored  by  finding  two  factors  of 
q  {the  absolute  term)  such  that  their  sum  is  p  {the  coefficient  of  x) 
and  adding  each  factor  of  q  to  x. 

Thus,  a^  +  8a?  +  15  =  (a;  +  S){x  +  5), 

x'-^x  +  lo^{x-  3)  (;r  -  5), 
a;2  4_  2  oj  -  15  =  (a:  -  3)(a;  +  5), 
x'-2x-lb  =  {x  +  3){x-^). 

EXERCISES 

155.  1.   Resolve  x~  —  13  ic  ~  48  into  two  binomial  factors. 

Solution.  —  The  first  terra  of  each  factor  is  evidently  x. 

Since  the  product  of  the  second  terms  of  the  two  binomial  factors  is 
—  48,  the  second  terms  must  have  opposite  signs ;  and  since  their  alge- 
braic sum,  —  13,  is  negative,  the  negative  term  must  be  numerically- 
larger  than  the  positive  term. 

The  two  factors  of  —  48  whose  sum  is  negative  may  be  1  and  —  48, 
2  and  —  24,  3  and  —  16,  4  and  —  12,  or  6  and  —  8.  Since  the  algebraic 
sum  of  3  and  —  16  is  —  13,  3  and  —  16  are  the  factors  of  —  48  sought. 


13x-48=  (x+3)(a:-16), 


2.    Factor  72 


Solution.  —  Arranging    the   trinomial   according   to   the   descending 

powers  of  m, 

72  -  m2  _  w»  =  -  m2  -  m  4-  72 

Making  m^  positive,  =  —  (m^  -f  m  —  72) 

=  -{m-  8)  (w  +  9) 

=  (-  m  +  8)  (m  +  9) 

=  (8  -  w)  (9  +  m). 


FACTORING  101 

Separate  into  simplest  factors  and  test  each  result  by  assign- 
ing a  numerical  value  to  each  letter  : 

3.  aj*  +  7«-l-12.  16.  a^-f 5aa;-f-6a*. 

4.  y*-7y-hl2.  17.  a^ - 6 ax -{- 5 aK 

5.  p^-Sp  +  12.  18.  f-4:by-12b', 

6.  r2  +  8r-|-12.  19.  /-3ny-28n*. 

7.  16  +  2a-al  20.  z--anz-2ahi\ 

8.  b-^b-12,  21.  x*  +  19cQ^-h90(^. 

9.  so-r^  +  r.  22.  a:* -f  12  oa^^ -f  20  a^ 

10.  c^-c- 72.  23.  a;'«-116V-f-24  6\ 

11.  c*  — 5  c  — 14.  24.  5  nx^  —  55  nx -\- 150  n. 

12.  a:*- a; -110.  25.  3 a^ftx^ _ 3  ^2^^ _ g ^2 ^ 

13.  -a*-9a4-52.  26.  4  aa;  4- 2  oa;' -  48  a. 

14.  a*H-8a-128.  27.  11  a^x- 55 ax  +  66a;. 

15.  _ar' -1-25  a; -100.  28.  20  6a; -|- 10  6*  -  630  «*. 

29.    Factor  ar^  -  (c -h  d)a; -I- cd. 

Suggestion.  —  Write  the  trinomial  in  the  standard  form, 

xi+(-c-d)x+  (-c)(-d). 

30.  Factora;*- (a-d)a;— ad.     31.  Factora;*— 2(a— n)a;— 4  an. 
156.   To  factor  a  trinomial  of  the  form  ax^  -\-bx  -^c. 

EXERCISES 

1.    Factor  3  a;*  4- 11  a; -4. 

Solution.  —  If  this  trinomial  is  the  product  of  two  binomial  factors, 
they  may  be  found  by  reversing  the  process  of  multiplication  illustrated 
in  exercise  .32,  page  68. 

Since  .3  z^  is  the  product  of  the  Jirst  terms  of  the  binomial  factors,  the 
first  terms,  each  containing  x,  are  3  x  and  x. 

Since  —  4  is  the  product  of  the  last  terms,  §  84,  they  must  have 
unlike  signs,  and  the  only  possible  last  terms  are  4  and  —  1,  —4  and  1,  or 
•_'  and  —  2. 


102  FACTORING 

Hence,  associating  these  pairs  of  factors  of  —  4  with  3  x  and  x  in  all 
possible  ways,  the  possible  binomial  factors  of  3cc^  +  11  x  —4  are  : 

3x  +  4\        3x-l\       3x-4)         3x+l-|         3X+2)         3x-2) 
x-li'       X+4J'       x+li"'        x-4/'       x-2/'       x  +  2r 

Of  these  we  select  hy  trial  the  pair  that  will  give  +  11  x  (the  middle 
term  of  the  given  trinomial)  for  the  algebraic  sum  of  the  "  cross-products," 
that  is,  the  second  pair. 

.-.  3x2  +  llx-4  =  (3x-l)  (x  +  4). 

Remark.  —  Since  changing  the  signs  of  two  factors  of  a  number  does 
not  change  the  value  of  the  number,  3x2  +  11  x  —  4  i^^s  also  the  factors 
(—  3x  +  l)  and  (-  x-4);  thus, 

3x2+llx-4=  (-3x  +  l)(-x-4). 

Such  negative  factors,  however,  are  not  usually  required. 

By  a  reversal  of  the  law  of  signs  for  multiplication  and  from 
the  above  solution  it  may  be  observed  that : 

1.  When  the  sign  of  the  last  term  of  the  trinomial  is  +,  the  last 
terms  of  the  factors  must  be  both  +  or  both  — ,  and  like  the  sign 
of  the  middle  term  of  the  trinomial. 

2.  When  the  sign  of  the  last  term  of  the  trinomial  is  —,  the  sign 
of  the  last  term  of  one  factor  must  be  + ,  and  of  the  other  — . 

Factor : 

2.  5x'-{-9x-2.  5.  3a^-7x-6. 

3.  2aj2-5aj-12.  6.  6a^-13a;  +  6. 

4.  3a^-17a;+10.  7.  6q^-11x-S5. 

When  the  coefficient  of  r^  is  a  square,  and  when  the  square 
root  of  the  coefficient  of  x^  is  exactly  contained  in  the  coefficient 
of  X,  the  trinomial  may  be  factored  as  follows : 

8.    Factor  9  a^  + 30  a; +  16. 

Solution 
9x2  +  30x4-16 

=  (3x)2  +  10(3x)  +  16 
=  (3x  +  2)(3x  +  8). 


FACTORING  103 

9.    Factor  4  a:*  — 5  a;  — 6. 

Solution 

4x»-6x-6  =  (4;r.-o»-6)xi  =  ""'-f^-'^ 

4  4 

_  (4g)a  -  6(4a;)  -  24  _  (4x  -  8)(4x  +  3) 
4  4 

=  ^(^-^H^^  +  ^)=(x-2)(4x4-3). 

Explanation.  —  Although  the  first  term  is  a  square,  its  square  root  is 
not  contained  exactly  in  the  second  term.  But  if  such  a  trinomial  is  mul- 
tiplied by  the  coefficient  of  x^,  the  resulting  trinomial  will  be  one  whose 
second  term  exactly  contains  the  square  root  of  its  first  term. 

Multiplying  the  given  trinomial  by  4,  factoring  as  in  exercise  8,  and 
dividing  the  result  by  4,  we  find  that  the  factors  of  the  given  trinomial 
are  (x  -  2)  and  (4  x  +  8). 

10.   Factor  24x^  +  Ux-5. 

Suggestion.  —  ]Mien  the  first  term  is  not  a  square,  it  may  always  be 
made  a  square  whose  square  root  will  be  contained  exactly  in  the  second 
term  by  multiplying  the  trinomial  by  the  coefficient  of  x^,  but  frequently  a 
smaller  multiplier  will  accomplish  the  same  result.  In  this  case  multiply 
by  6,  and  divide  by  the  same  number  to  avoid  changing  the  value  of  the 
expression. 

Separate  iuto  simplest  factors,  testing  results : 

11.  2a;*-fx-15.  21.   9a;*-10ar'-16. 

12.  9x»-42a:-f40.  22.    27b*-3b''-U. 

13.  5a*-f  13a;4-6.  23.    10 a;« - 2 af» - 44. 

14.  25a:«  +  16«4-2.  24.   2x^-^5xy +  2if. 
16.    IGx'-f 20a;-66.  26.   23e^-^3xy-2i/, 

16.  36a:*-48a;-20.  26.   3a^ -lOxy -{-Sf, 

17.  9x«-h43x-10.  27.   Wa^-Ux-S. 

18.  25  0^-1-25  a; -24.  28.    15  x« -f  17  a;  -  4. 

19.  49a^-42x-55.  29.    21a*-a-10. 

20.  16x*-|-50x-21.  30.    18ar'-3x-36. 


102  FACTORING 

Hence,  associating  these  pairs  of  factors  of  —  4  with  3  x  and  x  in  all 
possible  ways,  the  possible  binomial  factors  of  3a;^  +  11  x  —4  are  : 

3x  +  4l         3x-l\       3a: -4)         3x+l1         3a: +2")         3x-2) 
x-li'       X+4J'       x+li'        a:-4/'       a:-2J'       a: +  2) 

Of  these  we  select  hy  trial  the  pair  that  will  give  +  1 1  a:  (the  middle 
term  of  the  given  trinomial)  for  the  algebraic  sum  of  the  "  cross-products," 
that  is,  the  second  pair. 

.-.  3x2  +  11  a; -4  =  (3x-l)  (x  +  4). 

Remark.  —  Since  changing  the  signs  of  two  factors  of  a  number  does 
not  change  the  value  of  the  number,  3x^  +  11  x  —  4  has  also  the  factors 
(—  3x  +  l)  and  (-  x  — 4);  thus, 

3x2  +  llx-4=  (_3x  +  l)(-x-4). 

Such  negative  factors,  however,  are  not  usually  required. 

By  a  reversal  of  the  law  of  signs  for  multiplication  and  from 
the  above  solution  it  may  be  observed  that : 

1.  When  the  sign  of  the  last  term  of  the  trinomial  is  +,  the  last 
terms  of  the  factors  must  be  both  +  or  both  —,  and  like  the  sign 
of  the  middle  term  of  the  trinomial. 

2.  WJien  the  sign  of  the  last  term  of  the  trinomial  is  —,  the  sign 
of  the  last  term  of  one  factor  must  be  -f ,  and  of  the  other  — . 

Factor : 

2.  ^a?-\-^x-2.  5.  Zx'-lx-Q. 

3.  2x2_53._i2.  6.  6a^-13a!  +  6. 

4.  Zx^-llx^W.  7.  6a^-llic-35. 

When  the  coefficient  of  ar^  is  a  square,  and  when  the  square 
root  of  the  coefficient  of  x^  is  exactly  contained  in  the  coefficient 
of  X,  the  trinomial  may  be  factored  as  follows : 

8.    Factor  9  a^  + 30  a; +  16. 

Solution 
9x2  +  30x  +  16 

=  (3x)2  +  10(3x)  +  16 
=  (3x  +  2)(3x  +  8). 


FACTORING  103 

9.    Factor  4a^  — 5a;  — 6. 

Solution 

4x^— oaj  —  o  =  (4X'*  —  OX  —  ())  x-  = 

4  4 

^  (4g)a  -  6(4a;)  -  24  _  (4x  -  8)(4x  +  3) 
4  4 

=  iX?^^lKi^±^  =  (x-2)(4x  +  3). 

Explanation.  —  Although  the  first  term  is  a  square,  its  square  root  is 
not  contained  exactly  in  the  second  term.  But  if  such  a  trinomial  is  mul- 
tiplied by  the  coefficient  of  x'^,  the  resulting  trinomial  will  be  one  whose 
second  term  exactly  contains  the  square  root  of  its  first  term. 

Multiplying  the  given  trinomial  by  4,  factoring  as  in  exercise  8,  and 
dividing  the  result  by  4,  we  find  that  the  factors  of  the  given  trinomial 
are  (x  -  2)  and  (4  x  +  3). 

10.   Factor  24x^  +  Ux-5. 

Suggestion.  —  When  the  first  term  is  not  a  square^  it  may  always  be 
made  a  square  whose  square  root  will  be  contained  exactly  in  the  second 
term  by  multiplying  the  trinomial  by  the  coefficient  of  x^,  but  frequently  a 
smaller  nuiltiplier  will  accomplish  the  same  result.  In  this  case  multiply 
by  6,  and  divide  by  the  same  number  to  avoid  changing  the  value  of  the 
expression. 

Separate  into  simplest  factors,  testing  results : 

11.  2x2-fx-15.  21.    9a;*-10ar^-16. 

12.  9x*-42x  +  40.  22.    27b*-Sb^-U. 

13.  ^x'  +  lSx  +  e,  23.    10a^-2a^-U. 

14.  25a:»H-16x-f  2.  24.   2x'  +  5xy  +  2y'. 

15.  16iC*  +  20a;-66.  26.    2x'  +  3xy-2y'. 

16.  36a;'-48a;-20.  26.   Sx" -lOxy -{-Sy". 

17.  9x'-\-43x-10.  27.   Wx'-Ux-S. 

18.  25ar'-|-25x-24.  28.    15 «*  +  17 a; - 4. 

19.  49a;«-42a;-55.  29.    21a2_a-10. 

20.  16a:*-f  50x-21.  30.    18a:2_3a._36 


104  FACTORING 

157.    To  factor  the  sum  or  the  diiference  of  two  cubes. 
By  applying  the  principles  of  §§  134-136, 

n»  _1_  h^  ^3  _  7,3 

=^a'-ab-\-b^  and =  a^  +  ab  +  b\ 


a-\-b  a  —  b 

Then,  §  121,    a'  +  b'==(a-^  b)  (a'  -ab-{-  b'), 
and  a^-b^=(a-  b)  (a-  -\- ab  +  b'-). 

By  use  of  these  forms  any  expression  that  can  be  written  as 
the  sum  or  the  difference  of  two  cubes  may  be  factored. 

EXERCISES 

158.  1.  Factor  a:«  +  /. 

Solution 

X^-\-y^-^  (X^y  +  (y^)3  ^  (x-^  +1/2)  (x4  _  x''y2  +  y4^ . 

2.  Factor  a^-125b^ 

Solution 

a^  -  125  63  =  (a3)3  _  (5  5)3^  (^3  _  5  ^))  (^^e  4.  5  «3^  _|.  25  62). 

Factor,  and  test  each  result : 

3.  x^  +  f.  9.    x  —  x\  15.   r^^  — 729s'^. 

4.  ar^  — 2/^.  10.  v'-\-27v.  16.  512  a;*"» -f- 64  2/^. 

5.  m-^-1.  11.  o?W-(^dK  17.  1  +  (a  +  6)^ 

6.  1  +  ml  12.  r^  +  64  ^.  18.  (x  -  ?/)^  -  8. 

7.  x'-y''.  13.  0.^2:^-216.  19.  8  (?ri  +  ^0^  +  12o  ?il 

8.  r^  +  sl  14.  343  71^  +  1000.  20.  {x  -  yf  -  {x -{- y)\ 

159.  To  factor  the  sum  or  the  difference  of  the  same  odd  powers 
of  two  numbers. 

By  applying  §§  134-136,  as  in  §  157,  any  expression  that 
can  be  written  as  the  sum  or  the  difference  of  the  same  odd 
powers  of  two  numbers  may  be  resolved  into  two  factors.     Thus, 

a'  +  b'  =  (a  +  6)(a^  -  a^b  +  a'b^  -  ab^  +  b'), 
and  a'-b'  =  {a-  b)(a*  +  a^b  +  a-6-  +  ab^  +  b*). 


FACTORING  106 

EXERCISES 

160.  1.   Factor  m'^  4- 32  ar^. 
Solution.  —  rn*  +  32  ac*  =  m»  +  (2  a;)» 

2.  Factor  128  a" -1. 

Solution 
\2Sa^*-\=(2a'^y-\ 

=  (2a2-  l)(64a"  +  32ai'>  +  16a8  +  8a«4-4a*  +  2a2+  1). 
Factor : 

3.  m*  4-  w*.  8.1  +  a^  13.  mJ  —  mV. 

4.  7?i*-7i'.  9.   xf^-}f.  14.  a^b  —  ab\ 

5.  x^-1.  10.  ar''  +  y«.  15.  x^-y^, 

6.  ar»  +  /.  11.    a^  +  32.  16.  1  -  a*6' V\ 

7.  a^-x.  12.   64 -2 a*.  17.  x^«4-243a*. 

161.  To  factor  the  difference  of  the  same  even  powers  of  two 
numbers. 

EXERCISES 

1.   Factor  a«-M. 

First  Solution 
§§  134-136,  fl{«  _  66  =(a  -  6)(a*  +  a*6  +  a^f^  +  a^ft*  +  ab*  +  b^) 
=  (a  -  6)(a*  +  a26»  +  a*6  +  a6*  +  a'ft*  +  6«) 
=  (a-  b)[a^(a»  +  6«)  +  a6(a»  +  6«)  +  62(a«  +  f''^)] 
=  (a  -  6)  (a2  +  aft  +  ft^)  (a«  +  ft») 
§  167,  =  (o  -  *)(«*  +  a6  +  60 (a  +  b) (a^  -  ab -^  6^. 

Second  Solution 
§  152,  a«  -  6»  =  (a«)2  -  (fe»)2  =  (a»  +  6»)  (a»  -  6») 

§  157,  =  (a  +  6)(a2  _  a6  +  6«)(a  -  6)(a2  +  aft  +  ft^). 

Considering  the  even  powers  as  squares,  as  in  the  second  solution, 
the  process  may  be  regarded  as  factoring  the  difference  of  ttoo  squares. 


106  FACTORING 

Separate  into  simplest  factors : 

2.  x^-y^.                      5.    aj^-16.  ~           8.  1-6". 

3.  x^-1.                        6.    x^-^1.  9.  64-/. 

4.  a^-h\                      7.    a^-625.  10.  1-a^. 

162.  All  the  preceding  methods  of  finding  binomial  factors 
are  really  special  methods.  The  following  is  a  general  method 
of  finding  binomial  factors,  when  they  exist. 

163.  To  factor  by  the  factor  theorem. 

Zero  multiplied  by  any  number  is  equal  to  0. 

Conversely,  if  a  product  is  equal  to  zero,  at  least  one  of  the 
factors  must  be  0  or  a  number  equal  to  0. 

If  5  a.-  =  0,  since  5  is  not  equal  to  0,  x  must  equal  0. 

If  5(x  —  3)  =  0,  since  5  is  not  equal  to  0,  x  must  have  such 
a  value  as  to  make  x  —  3  equal  to  0 ;  that  is,  x=  3. 

If  5{x  —  d),  or  5a5  — 15,  or  any  other  polynomial  in  x  re- 
duces to  0  when  a;  =  3,  ic  — 3  is  a  factor  of  the  polynomial. 

Sometimes  a  polynomial  in  x  reduces  to  0  for  more  than  one 
value  of  X.     For  example,  x^  —5x-\-Q  equals  0  when  a;  =  3  and 
also  when  a?  =  2 ;    or  when  ic  —  3  =  0  and  a;  —  2  =  0.     In  this 
case  both  x  — 3  and  a;— 2  are  factors  of  the  polynomial. 
.•.aj2_5a;_|_6=(aj-3)(aj-2). 

164.  Factor  Theorem. — If  a  polynomial  in  x,  having  positive 
integral  exponents,  reduces  to  zero  ivhen  r  is  substituted  for  x,  the 
X)olynomial  is  exactly  divisible  by  x  —  r. 

The  letter  r  represents  any  number  that  we  may  substitute  for  x. 

Proof. —  Let  Z)  represent  any  rational  integral  expression  containing 
ic,  and  let  D  reduce  to  zero  when  r  is  substituted  for  x. 

It  is  to  be  proved  that  D  is  exactly  divisible  hy  x  —  r. 

Suppose  that  the  dividend  D  is  divided  by  x  —  r  until  the  remainder 
does  not  contain  x.     Denote  the  remainder  by  B  and  the  quotient  by  Q. 

Then,  D=  Q(x-r)+  B.  (1) 

But,  since  D  reduces  to  zero  when  x  =  r,  that  is,  when  x  —  r  =  0, 
(1)  becomes  0  =  0  +  B  ;   whence,  B  =0. 

That  is,  the  remainder  is  zero,  and  the  division  is  exact. 


FACTOKING  lOT 

BXBRCISBS 

165.   1.  Factor  ar»  -  ar^- 4  a;-f  4. 

Solution 
Whenx  =  l,        x'  — a;^-4a;  +  4  =  l-l-4  +  4=0. 
Therefore,  x  —  1  is  a  factor  of  the  given  polynomial. 
Dividing  x*  —  x^  —  4  x  +  4  by  x  —  1,  the  quotient  is  found  to  be  x*  —  4. 
By  §  152,  X"*  -  4  =(x  +  2)(x  -  2). 

...  a*  _  a;2  _  4a;  +  4  =(x  -  l)(x  +  2)(x  -  2). 

Sdggestions.  —  1.  Only  factors  of  the  absolute  term  of  the  polynomial 
need  be  substituted  for  x  in  seeking  factors  of  the  polynomial  of  the  form 
X  —  r,  for  if  X  —  r  is  one  factor,  the  absolute  term  of  the  polynomial  is 
the  product  of  r  and  the  absolute  term  of  the  other  factor. 

2.  In  substituting  the  factors  of  the  absolute  term,  try  them  in  order 
beginning  with  the  numerically  smallest. 

3.  When  1  is  substituted  for  x,  the  value  of  the  polynomial  is  equal  to 
the  sum  of  its  coefficients ;  then  x  — 1  is  a  factor  when  the  sum  of  the 
coefficients  is  equal  to  0. 

2.  Factor  17  ar^- 14x2 -37  a: -6. 

Solution 

Since  the  sum  of  the  coefficients  is  not  equal  to  0,  x  —  1  is  not  a  factor. 

When  X  =  -  1,  17  x-J  -  14  x2  -  37  X  -  6  =  -  17  -  14  +  37  -  6  =  0. 

Therefore,  x  —  (  —  1),  or  x  +  1,  is  a  factor  of  the  given  polynomial. 

Dividing  Hx*  —  Hx^  —  37x  —  6  by  x  +  1,  the  quotient  is  found  to  be 
17  X*  —  31 X  —  6,  which  in  turn  may  be  tested  for  factors  by  the  factor 
theorem. 

Substituting  factors  of  —  6  for  x,  it  is  found  that : 

Whenx  =  2,       Hx^  -  31  x  -  6  =  68  -  62- 6  =0. 

Therefore,  x  —  2  is  a  factor  of  17  x^  —  31  x  —  6. 

Dividing  by  x  —  2,  the  other  factor  is  found  to  be  17  x  +  3. 

.-.  17x'  -  14x2  -  37x  -  6  =(x  +  l)(x  -  2)C17x  +  3). 

3.  Factor  2ar»  +  a^  — 5  a:/ +2^*. 
Suggestion.  —  When  x  =  y, 

2  x»  +  x2y  -  5  xy2  -I-  2  y»  =  2  y«  +  y«  —  5  y«  +  2  y«  =  0. 
Therefore,  x  —  y  is  a  factor  of  2x»  +  x^y  —  5xy*  +  2 y«. 


108  FACTORING 

Factor  by  the  factor  theorem  : 

4.  x'-Slx-^SO.  24.  a^-67x-126. 

5.  4a^-7a;  +  3.  25.  a;3_39a;_70. 

6.  26  0.-2 -10  a; -16.  26.  a'^  +  4a- -  11  a-30. 

7.  48a^-31a;-17.  27.  a^ -{- 9  a^ -}- 26  a  +  24:. 

8.  36a^-61a;  +  25.  28.  m^-6m--m  +  30. 

9.  a,'3-9a^  +  23.T-15.  29.   b^-5b^ -29b-^10D. 

10.  a^-13aj2-i-47a:-35.  30.  a^^  lOa-'-^a- 66. 

11.  x^-14:X^  +  35x-22.  31.   m3  +  7??i2-f 2m-40. 

12.  a^-4a^-7aj  +  10.  32.   5^  + 16  6^  _|_  73  6  +  90. 

13.  x'-Qx^-dx-^-U.  33.   n34-12n2  +  41?i  +  42. 

14.  a.-3_12a^  +  41aj-30.  34.  a;^  - 15 x^  + 10 a;  +  24. 

15.  a?3-lla^  +  31a?-21.  35.  a.'^  -  25  a;-' +  60  aj  -  36. 

16.  ar^-10.a^H-29a;-20.  36.  a.-^  +  13  a;^  -  54  a;  +  40. 

17.  ar^-16a;2  +  71a;-56.  37.  a;*  +  22 a.-^  +  27 « -  50. 

18.  a^-57a7  4-56.  38.   x*  -  9  a^y- -  4.  xf -\- 12  y*. 

19.  ar^-21aj/  +  20^.  39.   x'-9xY +  12 xf-4:y\ 

20.  x^-31xy~-S0f.  40.   a;^  _a^- 7a;2_^^_^6^ 

21.  x^-13xy^-\-12f.  41.   a;^-9a;3  +  21a;2_,_^_3()^ 

22.  a^  — 7a;4-6.  42.   a;*  +  8x^H-14a;2  — 8a;— 15. 

23.  ar^-19a;  +  30.  43.   a:^- 4a^*  + 19^^- 28a;  + 12. 

44.  a;^-18a;»4-30a^-19a;H-30. 

45.  a;^-10a;^  +  40ar^-80a;2  +  80a;-32. 


FACTO  RIXC;  109 

SPECIAL   APPLICATIONS   AND    DEVICES 

166.  Factor: 

1.  a--]-b-^c'-\-(P-\-2ab-2ac-\-2ad-2bc-{-2bd-2cd. 

Solution.  — Since  the  polynomial  consists  of  the  squares  of  four  num- 
bers together  with  twice  the  product  of  each  of  them  by  each  succeeding 
number,  the  polynomial  is  the  square  of  the  sum  of  four  numbers,  §  111, 
and  may  be  separated  into  two  ecjual  factors  containing  «,  6,  c,  and  d 
with  proper  signs. 

Since  the  a6,  ad^  and  bd  terms  are  positive,  a  and  6,  a  and  d,  and  6 
and  d  must  have  like  signs ;  since  the  a<*,  he,  and  rd  terms  are  negative^ 
n  and  c,  h  and  c,  and  c  and  d  must  have  unlike  signs. 

Therefore,  the  factors  are  either 

(a  +  6-C+  d)(a  +  b-c  +  d) 
or  (— a -6  +  c  — d)(— a  — 6  +  c  — d). 

2.  9x*-^Af-^25z^-12xy-^30xz-20yz. 

3.  25  m^  +  36  ?i2  4-^2  _  60  7WW  -  10  7np  -f- 12  ?*;). 

4.  a'^-\-16x^  +  36f-Sax^  +  12ay-'iS3^y. 

5.  a:2^4a2_,_^2_|_y2^4^_2^j;4.2a;//-4a6H-4ay-26?/. 

6.  m'-|-4  «2  +  02  +  9  —  4  ?/i/i  —  2  am  +  6  //i  +  4  cut  —12h-6  a. 

167.  The  principle  by  which  the  difference  of  two  squares 
is  factored  has  many  special  applications. 

1.  Factor  a*  +  0*62  ^6\ 

SoLDTioN.  —  Since  cr*  +  a'^h^  +  ft*  lacks  +  a-b'^  of  being  a  perfect  square, 
and  since  the  value  of  the  polynomial  will  not  be  changed  by  adding  a-b- 
and  also  subtracting  a^ft*,  the  polynomial  may  be  written 

•    a*  +  2a^b^  +  b*-aib'^, 

which  is  the  difference  of  two  squares. 

.-.  a*  +  a^b^  +  6*  =  a*  +  2  a'^ft*  +  6<  -  a^b^ 
=  (a^+b'iy-a^b^ 
=  (a-*  +  a6  +  62)  (^2  -  ab  +  f^). 

2.  Factor  4a*-13rt«  +  9. 

Suggestion.      4  a*  -  13  a*  +  9  =  4  a*  -  12  a^  +  9  -  a^  =  (2  a*  -  3)2  -  a^. 


110  FACTORING 

3.  Factor  a*  +  4. 

Suggestion.      a*  +  4  =  a^  +4  a^  +  4  —  4  a^  =  (a^  +  2)2  -  4  a^. 
Factor  the  following : 

4.  X* -\- o^y- -ir  y\  10.    ic^  +  a^-fl. 

5.  a'^  +  a^fe^  +  ftl  11.  n»H-n*4-l. 

6.  9.t4  +  20x2/  +  16  2/^  12.  16a;^H-4afy4-?/^ 

7.  4.a'-\-lla'b^  +  9b\  13.  a^6* -  21  «-&2  +  36. 

8.  IGa^-lTa^a^  +  ojl  14.  25  a' -  U  a'b^ -\- b\ 

9.  25«^-29a^/  +  4?/^  15.  9  a' -\- 26  a'b^ -\- 25  b*. 

16.  6* +  64.  19.    a^  +  324.  22.    a;*  +  64y*. 

17.  a^  +  4  6^  20.    a«-16.  23.    4a^-f81. 

18.  m^  +  4.  21.    m^-{'4:mn\  24.    ary  +  4a:/. 

168.    Many  polynomials  may  be  written  in  the  form  x^-\-px 
-\-  q,  X-  and  x  being  replaced  by  polynomials. 

1.  Factor  9  a;-  +  4  /  +  12  z-  +  21xz-\-  14  yz  + 12  xy. 

Solution.  9  x^  -^  4:1/^  +  \2  z^  +  2\  xz  +  Uyz  +  12  xy 

=  (9  x^  +  12  a:^  +  4  ^2)  +  (21  x^  +  14  yz)  +  12  ^^ 

=  (.3  X  +  2  ?/)2  +  7  0(3  X  +  2  y)  +  4  ^  .  3  2 
§154,  ={Sx  +  2y  +  4:z)(Sx-{-2y+Sz). 

Factor  the  following : 

2.  a'-i-2ab-\-b^-^Sac-\-Sbc+15c\ 

3.  a^  —  6  a??/  -f  9  ?/-  +  6  x'z  —  18  2/2;  4-  5  z',. 

4.  m^  4-  '^*^^  —  2  m^i  +  7  mp  —  7  np  —  30  p'-. 

5.  16  n^  + 55 —  64  w  — 16  m  +  m^  + 8  wi^i. 

6.  9m^  +  A:2-30  +  39m2  +  13A:  +  6m%. 

7.  25a-  +  ?/2  +  10x2H-10a?/-35a:»-7a;?/. 

8.  a^  +  b-  +  c-  +  2ab+2ac  +  2bc  +  5a  +  5b  +  oc  +  6. 


FACTORING  111 

REVIEW  OF  FACTORING 

169.    Summary  of  Cases.  —  In  the  previous  pages  the  student 
has  learned  to  factor  expressions  of  the  following  types : 

I.   Monomials ;  as  a^b^c.  (§  144) 

II.    Polynomials  whose  terms  have  a  common  factor ;  as 

nx -\-n/ -\- nz.  (§140) 

III.  Polynomials  whose  terms  may  be  grouped  to  show  a  com- 
mon polynomial  factor ;  as 

ax  +  ay  -\-  bx  -{-  by. 

IV.  Trinomials  that  are  perfect  squares ;  as 

a^-h  2ab  +  b^z,ii(}id'-2ab  +  b\ 
V.    Polynomials  that  are  perfect  squares ;  as 
a-  +  b-  4-  c-'  +  2  a6  -h  2  ac  +  2  be. 
VI.   The  difference  of  two  squares ;  as 


(§  147) 

(§§ 

148-151) 

(§  166) 

(§  152) 

(§  167) 

(§  154) 

(§  156) 

and  a'  +  d'b^  +  b\ 

VII.    Trinomials  of  the  form 

x'+px  +  q. 

VIII.   Trinomials  of  the  form 

ax'  -\-bx-{-c 

IX.   The  sum  or  the  difference  of  two  cubes ;  as 

a''  +  6"'ora-^-6\  (§  157) 

X.   The  sum  or  the  difference  of  the  same  odd  powers  of  two 

numbers:  as 

a"  +  A"  or  a"  -  6"  (when  n  is  odd).     (§  lo9) 

XI.    The  difference  of  the  same  even  powers  of  two  numbers ;  as 

fl»  _  ^w  (when  n  is  even).  (§  161) 

XII.    Polynomials  having  binomial  factors.  (§§  162-165) 


112  FACTORING 

170.  General  Directions  for  Factoring  Polynomials.  —  1.   Re- 
move monomial  factors  if  there  are  any. 

2.  Then  endeavor  to  bring  the  polynomial  under  some  one  of 
the  cases  II- XL 

3.  When  other  methods  fail,  try  the  factor  theorem. 

4.  Resolve  into  prime  factors. 

Each  factor  should  be  divided  out  of  the  given  expression  as  soon  as 
found  in  order  to  simplify  the  discovery  of  the  remaining  factors. 

171.  Factor  the  following : 

1.  2/^—1.  9.  y  —  a*y.  17.  8  — 27aV. 

2.  1-a^.  10.  x'y-f.  18.  32ic-2a^. 

3.  a;iO-l.  11.  a^^-ab^\  19.  6  6^  +  24. 

4.  x^-1.  12.  a* -256.  20.  a'  +  27aK 

5.  a-a\  13.  64. -2  f.  21.  b^'-lde. 

6.  b'-\-b.  14.  1  n'' —  1  n.  22.  450  — 2  a^ 

7.  ^4_^4.  15.  4  a;*  — 4  a;.  23.  4m^  +  .004. 

8.  l+a;'^  16.  7  2/' -175.  24.  125-8a;«. 

25.  a;2  -  a;^  - 132 /.  36.  x'-ax-12aK 

26.  ax-  —  3  aa;  —  4  a.  37.  n^  —  an  —  90  a^. 

27.  ar' +  5  a;2  —  6  a;.  38.  a^b'^ab  —  ^6. 

28.  3a;2  +  30x  +  27.  39.  10  a^c  +  33  ac-7c. 

29.  128  a'^- 250  a^  40.  60  ti?/^  -  61 713/ -  56  n. 

30.  Sajio  +  lOa.'^-lS.  41.  2b  a?  +  m  xy  +  S6  y\ 

31.  6ar^-19a;  +  15.  42.  6  ax^  +  b  aayy  —  6  ay\ 

32.  aj^"  4-  2  x^'y^  +  2/^^.  43.  169  x^  -  26  aar'  +  aV. 

33.  7a;2_77^^_34^,  44.  aV  +  a^^'c^+fe^ 

34.  2/'-25  2/a;  +  136a:2^  45.  16  a;*  +  4  a^y  +  3/^ 

35.  9  ar^- 24  a;?/ +  16/.  46.  6V- - 13  ft^c  +  42  c. 


FACTORING  113 

47.  17x2^25aj-18.  62.  x'  +  x'y- 41xf-105y'. 

48.  5a^-26xy  +  5y2.  63.  x" -cx  +  2 dx-2cd. 

49.  f-{-16ay-36a\  64.  x^y -\- 4 ary -31  xy- 70 y. 

60.  8a«-21a6-96«.  65.  x^ -3 ax -\- 4: bx- 12 ab. 

61.  60o»  +  8ax-3a:2  gg    ctx^-9ax^-\-26ax-2'ia. 
52.  30«*-37x-77.  67.  12aa;-86a;-9ay+66y. 
63.  2  a^  +  28  a^H- 66 a:.  68.  25a:*-9y*-24yz-162*. 
54.  a«  +  6*-c2-2a6.  69.  ar'-224./_a«-2a^4-2az. 
66.  ox*  4- 10  ox -39  a.  70.  2b^m-3ab^-h2bmx-3abx. 

66.  7i*  +  >iV6*  +  a^6«.  71.  a^-\-b^-\-c'-2ab--2ac+2bc. 

67.  aV  +  aV-|-a«.  72.  ar^i/^- 14a^y -|-43a^  +  30y. 

68.  a*-16a-17.  73.  a:»y- 15x2|/  +  38a:y-242/. 

69.  a«a^-4aa;  +  3.  74.  ab3^-\-3aba^ -abx-3  ab. 

60.  6«-f.6y  +  y*.  75.  3bnix  +  2bm  —  3anx  —  2an, 

61.  af-2a^4-a;.  76.   20aar'- 28 aa^  + 5 a*a;- 7 a«. 

77.  «=^  +  92/*  +  25z*-6an/-10a:z  +  302/2. 

78.  9ar^  +  y«-f  16z2_6an/-82/z4-24a!z. 

79.  x'fz'  +  a^b^  +1  +  2  tt6x?/2  +  2  a^z  +  2  a6. 

80.  a'b^-^lf'c'  +  c»cP-2ab'c-h2abcd-2b(^d. 

81.  a:«  +  nV  +  n»  +  2wV  +  2nV+2nV. 

82.  a«6V-a262-6V+62-aV  +  a2  +  a:*-l. 

83.  (a  +  6)«-l.  88.    3a:«  +  96x. 

84.  a*_2tt*  +  l.  89.    (a-2y-\-(a-iy. 
86.    6»--46«  +  8.  90.    12a:'  +  3a^-8a;-2. 

86.  a;»-10a^  +  125.  91.   2a:»  +  10a:  +  aa;  +  5 a. 

87.  8aJ*-6a;*-36.  92.   a:»  +  5 a;* - 29 a; - 105. 

]fn.NE*»    8TAXD.    ALO. — 8 


114  FACTORING 

93.  a-b^-'iahx-4:X  +  2ab  +  4:a^. 

94.  (a  +  by(x-y)-{a  +  b)(x'-y^). 

95.  1-x^-^abx^-j-bx^  —  bx-ab. 

96.  x^  —  x^  4-  x-y  —  xy  -{- x^y  —  xy^. 

97.  x^^'-'  +  bY  -\-2x''-'^by. 

98.  x^-\-15x^-^75x-i-125, 

99.  4.(ab-{-cd:f-(a'  +  b--c^-ciy. 

100.  a^3n_^35  ;m  a;'' +  4  a;. 

101.  (a2  +  &2 _  e-y  _ 4  ^2^2^  -LJ2^  a^_ar^_.^4_,_^^ 

102.  a^62^o-6-12.  113.  (a  +  6)4  _  (6  _  c)4. 

103.  x^  —  xy  -  x^y -]- y\  114.  3  aZ>(rt  +  6)  +  a^  + 6^ 

104.  x^-4x2/  +  2cc-*'-16?/^  115.  (x-\-yf+{x-yy, 

105.  a^-64_(<^_l_^)(-^_^)^  116.  d'-i^a  +  by. 

106.  ar^  -  6  a;2  + 12  a; -8.  117.  x^- 119  ar^^/S  +  ^Z^ 

107.  1000  a^- 27  2/^.  118.  ?7i^+ m^ -mii -miil 

108.  {a  +  xy-x\  119.  (a;=^-/)--(a;--a!?/)l 

109.  l  +  (a^  +  l)3.  120.  x^-y^-SaryXx^-y"). 

110.  a5  —  ^a;"  +  a^Y"  —  «^"'.  121.  (x^-\-6x-{-9y-{x^+ox-\-6y. 

122.  2-3&  +  3a6-2a4-4a2-6a26. 

123.  Factor  32  —  a^  by  the  factor  theorem. 

124.  Factor  16  +  5a5  —  lla;^  by  the  factor  theorem. 

125.  If  n  is  odd,  factor  x"  — a"  by  the  factor  theorem. 

126.  If  n  is  odd,  factor  a?"  +  r"  by  the  factor  theorem. 

127.  Factor  a;^  -  6  fta^^  + 12  d^^;  _  8  53  by  the  factor  theorem. 

128.  Discover  by  the  factor  theorem  for  what  values  of  w, 
between  1  and  20,  a;"  +  a"  has  no  binomial  factors. 


or 


FACTORING  115 

EQUATIONS  SOLVED  BY   FACTORING 

172.    1.   Find  the  values  of  x  in  a;^  -h  1  =  10. 

Explanation.  —  On    transposing 

FIRST  PROCESS  ^^le  known  term    1    to   the    second 

ar*  -f- 1  =  10  member,  the  first  member  contains 

2j2_.    9  the    second    power,    only,    of    the 

o    o  .      o  unknown  number.    On   separating 

*  each  member  into  two  equal  factors, 

z  '  X  =  o  '  o  ora;«a5  =  —  .■>•—  o. 
.'.  x=z  ±3  Si„ce,  if  X  =  3,  X .  X  =  3  .  3,  and  if 

r  =  -  3,  X .  X  =  -  3 .  -  3,  the  value  of  x  that  makes  x'^  =  9,  or  that  makes 
^-  +  1  =  10,  is  either  4-  3  or  -3  ;  that  is,  x  =  ±  3. 

Find  the  two  values  of  x  in  each  of  the  following : 

2.  ^2 4-8  =  28.  6.   a^-^   3=    84. 

3.  ar  +  l=50.  7.   x'-24:  =  120. 

4.  2^-5  =  59.  8.   ar'  +  ll  =  180. 

5.  x2-7  =  29.  9.   x2-ll  =  110. 

10.    Find  the  values  of  x  in  ar^  +  1  =  10. 

SECOND  PROCESS  Explanation.  —  The  first  process 

-      ^  _  ^^  is  given  in  exercise  1. 

""  In  the  second  process,  all  terms 

^  —  '^  =    ^  are   brought  to   the    first    member, 

(x  —  S)(x -{- 3)  =    0  which  is  factored  as  the  difference 

.•.  X  —  3  =  0,  whence  x  =  3  of  the  squares  of  two  numbers. 

or  x  +  3  =  0,  whence  x  =  —  3  Since  the  product  of  the  two  fac- 

.   3,  _.   .  3  tors  is  0,  one  of  them  is  equal  to  0. 

Therefore,   x  —  3  =  0  or  x  +  3  =  0; 

whence,  x  =  8orx=  — 3;  that  is,  x  =  ±  3. 

Solve  for  x,  and  verify  results : 

11.  x*  +  35  =  39.  14.  .r-- 31^  =  0. 

12.  x2-50  =  50.  15.  .n-4b^  =  0. 

13.  x2-f90  =  91.  .  16.  x--97r  =  0. 


116  FACTORING 

17.  a^-21=4.  22.  32-a^  =  28. 

18.  x'--56  =  H.  23.  65-^2  =  16. 

19.  x^-3a^  =  6a\  24.  4.x--8b^  =  Sb\ 

20.  x'-h5b'  =  6b*.  25.  a;2_^.25  =  25  +  m2. 

21.  ar^- 40  =  24.  26.  x" -S0  =  2(2b^ -V,). 

27.  Solve  x^  -j-  2  am  =  a^  4-  m^. 

Solution 
x2  +  2  awi  =  «'^  +  m2. 

x^  =  a"2  —  2  «w  +  m^. 
aj  •  X  =  (a  —  w)  (a  —  m) 
or  x  •  a;  =  —  (a  —  m)  —  (a  —  m). 

.-.  X  =  ±  (a  —  m). 
Solve  for  x,  and  verify  : 

28.  x''-c'  =  d'-2cd.  34.    x^- 0^  =  36  -  12  c. 

29.  ar'_62  =  46c  +  4c-.  35.    x^- 4  62  =  36- 24  6. 

30.  .^2-^2  =  6  71 +  9.  36.    x^-a^  =  9-6a. 

31.  a;2-f  10a  =  rt-  +  25.  37.    or -6^  =  4  —  4  61 

32.  ;c2-a2  =  2a  +  l.  38.    a:2_  ^252^2  a6  +  1. 

33.  x^-m'  =  Sm  +  16.  39.    x" -r'  =  b'-2  7%'. 
40.  Find  the  values  of  a:  in  ar^  4-  4  ic  =  45. 

FIRST    PROCESS  SECOND    PROCESS 

a.'^  -f  4  aj  =  45  ar  +  4  a;  =  45 

ar^  +  4a;-45=0  x^  +  4:X  +  'i  =  49 

(x-5)(x  +  9)=    0  (a;  +  2)(.c  +  2)  =  7.7or-7.- 
.-.  a;  — 5  =  0  or  a^  +  9=    0  .-.  a;  + 2  =  7  or  -  7 

_^-.  cc  =  o  or  —  9 


FACTORING  117 

KxFLANATioN.  — FoF  the  fifst  process  the  explanation  is  similar  to  that 
given  for  exercise  10. 

In  the  second  process,  it  is  seen  that,  by  adding  4  to  each  member  of 
the  equation,  the  first  uieiuber  will  become  the  square  of  the  binomial 
(2+2).  On  solving  for  (ac+2)  as  for  x  in  previous  exercises,  2+2=  ±7; 
whence,  2  =  ±  7  —  2  =  +  7  —  2  or  —  7  —  2  =  o  or  —  i). 

SroGESTio.v.  —  In  the  following  exercises,  when  the  coefficient  of  the 
fii-st  power  of  the  unknown  number  is  even,  either  of  the  above  processes 
may  be  used ;  but  when  it  is  odd^  the  first  process  is  simpler. 

Solve,  and  verify  results : 

41.  a^-6a;  =  40.  56.  f-\-^2  =  13y. 

42.  «2-8x  =  48.  57.  t-  +  63=:16t. 

43.  x^-5x=-4:.  58.  ^--60  =  11^. 

44.  x^-{-4:X-\-S  =  0.  59.  xr-7x=:lS, 

45.  ?-2  4-6r4-8  =  0.  60.  ar  +  10x  =  56. 

46.  ar-9a;  +  20  =  0.  61.  a:^ _^  12 a;  =  28. 

47.  aT^-3x  =  iO.  62.  n^  +  ll  71 +  30  =  0. 

48.  a:*-9a;  =  36.  63.  a:*  +  x -  132  =  0. 

49.  x2-flla;  =  26.  64.  32  =  4w  +  wK 

50.  a:*-12x  =  45.  65.  3s  =  SS-s^. 

51.  y2_i5y^54  gg  iCyO^x'-ex. 

52.  ir-21.y  =  46.  67.  4.v  =  /-192. 

53.  a:*-10a;  =  96.  68.  600  =  f-10y. 

54.  /-20y=96.  69.  c2  +  16c-36  =  0. 

55.  3r'-|-12y  =  85.  70.  P-{-Wl-S4  =  0. 

Solve  for  a:,  y,  or  z,  and  verify  results : 

71.  x'-^iibx  +  f/^O.  73.    a^-(a-{-b)x-^ab  =  0. 

72.  22  +  4a2-|-4a*  =  0.  74.    a^+-(c-f-d)x  +  cd  =  0. 


118  FACTORING 

75.  ocf^-\-{a-\-2)x-\-2a  =  0.  77.   a^- (a-d)x-ad  =  0. 

76.  y^-(c-n)y-nc=0.  78.    x^  -  (b -^7)x +  7  b  =  0, 

79.  (2x-\-3)(2x-^)-(3x-l)(x-2)  =  l. 

80.  (2a;-6)(3a;-2)-(5a;-9)(a;-2)  =  4. 

81.  Solve  6a^  +  5a;-21  =  0. 

Solution 
6a:2  +  5a;-21  =  0. 
Factoring,  §  156,  (2  x  -  3)  (3  a;  4-  7)  =  0. 

.•.2a;-3  =  0 
or  3  a;  +  7  =  0. 

.•.x  =  for-|. 
Solve,  and  verify  results : 

82.  3ar'  +  2a;-l=0.  87.    7x^  +  6x-l  =  0, 

83.  5a^  +  4ic-l=0.  88.    2  v^-g  v_35  ==o. 

84.  3f  +  y-10  =  0.  89.    6^- 22^  +  20  =  0. 

85.  3/-42/-4  =  0.  90.   3.T2  +  13a;-30  =  0. 

86.  42/'  +  92/-9  =  0.  91.   4a;2  +  13x-12  =  0. 

92.  Solve  the  equation  oi:^  —  2xr  —  5x-\-6  =  0. 

Solution 

a:3  _  2  x2  -  6  X  +  6  =  0. 
Factoring,  §  163,  (x  -  1)  (x  -  3)  (a:  +  2)  =  0. 

.-.  x-l=0  orx-3  =  0  orx+2  =  0; 
whence,  x  =  1  or  3  or  —  2. 

93.  x^-Wx'-{-71x-105  =  0.  95.   a^ -12a;  + 16  =  0. 

94.  a-«  +  10x2  +  llaj-70  =  0.  96.    ar^- 19a;-30  =  0. 

97.  .T^-|-a:3-21.T2-a;  +  20  =  0. 

98.  x'-7x''-\-x'-^63x-90  =  0. 

99.  a!^-llx4  +  45ar'^-8oar  +  74a;-24  =  0. 


HIGHEST   COiMMON   FACTOR 


173.  The  sum  of  the  exponents  of  the  literal  factors  of  a 
rational  integral  term  determines  the  degree  of  the  term. 

Thus,  a  and  5  a  are  of  the  first  degree  ;  3  x^  and  3  xy  are  of  the  second 
degree  ;  4  ahh  and  x^(y  —  1)*  are  of  the  fifth  degree. 

174.  The  term  of  highest  degree  in  any  rational  integral 
expression  determines  the  degree  of  the  expression. 

Thus,  the  expression  a:'  —  Cx^  +  llx  —  6isof  the  thu-d  degree. 

175.  An  expression  that  is  a  factor  of  each  of  two  or  more 
expressions  is  called  a  common  factor  of  them. 

176.  The  common  factor  of  two  or  more  expressions  that 
has  the  largest  numerical  coefficient  and  is  of  the  highest 
degree  is  called  their  highest  common  factor  (H.  C.  F.). 

The  common  factors  of  4  a^b-  and  6  a'^b  are  2,  a,  b,  cfi,  2  a,  2  6,  2  a*,  aft, 
2  aft,  a'ft,  and  2  a^ft,  with  sign  +  or  - .  Of  these,  2  aVj  (or  —  2  a^b)  has 
the  largest  numerical  coefficient  and  is  of  the  highest  degree,  and  is  there- 
fore the  highest  common  factor. 

The  highest  common  factor  may  be  either  positive  or  negative,  but 
usually  only  the  positive  sign  is  taken. 

The  highest  common  factor,  or  divisor,  of  4  a^b'^  and  6  a^ft  is  2  a^ft, 
regardless  of  the  values  that  n  and  ft  may  represent.  What  the  arith- 
metical greatest  common  divisor  is  depends  upon  the  values  of  a  and  ft. 
If  a  =  2  and  ft  =  6, 

H.  C.  F.  =  2  a^ft  =  48  ;  but  since  4  a'ft^  =  1162  and  6  a«ft  =  144,  the 
arithmetical  greatest  common  divisor  =  144. 

177.  Principle.  —  The  highest  common  factor  of  two  or  mx)re 
expressiojis  is  equal  to  the  product  of  all  their  common  prime 
factors. 

178.  Expressions  that  have  no  common  prime  factor,  except 
1,  are  said  to  be  prime  to  each  other. 

119 


120  HIGHEST   COMMON   FACTOR 

EXERCISES 

179.    1.   Find  the  H.  C.  F.  of  12  a'b'c  and  32  a'b^(^. 
Solution 

The  arithmetical  greatest  common  divisor  or  highest  common  factor 
of  12  and  32  is  4.  The  highest  common  factor  of  a'^h^c  and  a^b-^c^  is  a'^b'^c. 
Hence,  H.  C.  F.  =  4  a^b'^c. 

KuLE.  —  To  the  greatest  common  divisor  of  the  numerical 
coefficients  annex  each  common  literal  factor  with  the  least  expo- 
nent it  has  in  any  of  the  eoepressions. 

Find  the  highest  common  factor  of : 

2.  10  a^y^  10  x^f,  and  15  xy^z, 

3.  70  a«6^  21  a%\  and  35  a^5l 

4.  8  m^n\  28  m^n\  and  56  m^n\ 

5.  4  hhd,  6  6V,  and  24  aU^. 

6.  3(a  +  6)2  and  6(a  +  6)1 

7.  6(a  +  6)2  and  4(a  + 6) (a -6). 

8.  12(a  -  xf,  Q{a  -  xf,  and  (a  -  x)\ 

9.  30(a;  +  yf,  18(a^  +  y),  and  (x  +  yf. 

10.  10(a;  -  2/) V  and  15(z  -  y)(x  -  yf. 

11.  3(a2  -  by  and  a{a  -  6)(a2  -  6'0. 

12.  What  is  the  H.C.F.  of  3  a:^-3a;/  and  6a;^-12  a^?/+6a;/? 

PROCESS 

3a^— 3a;?/2  =iZx{x-\-y){x  —  y) 

6a^-12x^y  +  6xy^  =  2'Sx(x-y)(x-y) 
.-.  B..C.¥.  =  3x(x-y) 

Explanation.  —  For  convenience  in  selecting  the  common  factors,  the 
expressions  are  resolved  into  their  simplest  factors. 

Since  the  only  common  prime  factors  are  3,  x,  and  (x  —  y) ,  the  highest 
common  factor  sought  (§  177)  is  their  product,  3  a:  (x  —  y). 


HIGHEST  COMMON   FACTOR  121 

Find  the  highest  common  factor  of : 

13.  ie*-2a:-16anda:^-ir-20. 

14.  ar*  —  y*,  ic*  —  y^  and  x-\-y. 

15.  a*  +  7  a  -f  12  and  o*  -h  5  a  +  6. 

16.  ar'  +  y*  and  «=  +  2ajy +  2/*. 

17.  a*  —  x' and  a' —  2  aa; -f  aj*. 

18.  a^-  I)'  and  a^  -f-  2  a6  -|-  b*. 

19.  ar*  +  ar*y^  4- y^  and  a?*  +  ary  +  ^. 

20.  af*  +  y»,  a^  4-  /,  and  x^y  +  xif. 

21.  a*  +  a'6*  +  6'and3a*-3a6*  +  36*. 

22.  a*  —  aj*,  a*  -f  2  ax  4-  a^,  and  a'  +  x*. 

23.  dx  —  y  +  xy  —  a  and  ani^ -\- x^y  —  a  —  y. 

24.  a^b  —  b  —  ah  +  c  and  a6  —  ac  —  6  +  c. 

25.  l-4a:*,  l  +  2a;,  and4a  — IGoa:*. 

26.  (a  -  b)  (b  -  c)  and  (c  -  a)  (a^  -b^, 

27.  24x»/4-8a^y'and8xy-8ar'y'. 

28.  6 x*  +  a;-  2  and  2 a^  — 11  x  +  5. 

29.  16x*-25and20x*-9x-20. 

30.  X*  -I-  xy*  and  x^y  +  x^. 

31.  pq*  +  p*7  and  qp^  +  ^^p*. 

32.  17  abc^d'  -  51  a»6c*d*  and  abc'(P  -  3  a^ftc^d. 

33.  38  xyz  -  95  x»t/2*  and  34  xfz  -  85  a^yz\ 
84.  x'y  4- a^  and  2  x*y  —  2  xy  +  2  xy*. 

36.  6i^4-10y»«-4r'«*and2r'4-2r««-4?^««. 

36.  x«-x*-2x*,  x*-2x»-3x*,  andx*-3x»-4x«. 

37.  3  m'w*  —  3  win*  and  6  m*n^  4-  6  m^n*  —  6  m*rv^  —  6  7?in*. 

38.  7  P^  +  35  /2^  4-  42  /<»  and  7  ?*<«  4-  21  f*/^  -  28  Pf"  -  84  /^. 

39.  aJ*4-a*-6«4-2ax,  x2-a24-6«4-26x,and«*-(i'-(''-2(^. 


122  HIGHEST   COMMON   FACTOR 

Apply  the  factor  theorem  when  necessary. 

40.  a^-6x  +  5  Sindx^-5x^-\-7x-3. 

41.  0^2-4  and  a^-10a^  +  31a;-30. 

42.  a;3-4a;  +  3andaf'  +  aj2-37a;  +  35. 

43.  Sx*-12a^  Sind6x'-\-S0a^-96x'-^24:X. 

44.  a*b  -  a^b^  and  a*b  +  2  a^b^  +  2  a^ft^  ^  ab*. 

45.  9  —  71^  and  w^  —  ti  —  6. 

Suggestion.  —  Change  9  -  w^  to  —  (^2  _  9)  :=  _  (^  +  3)  (,1  _  3). 

46.  1-a^and  a^-6a^-9aj  +  14. 

47.  4-a2anda*H-a3-10a--4a  +  24. 

48.  (9-a;2)2anda;^  +  5a:3-3a^-45a;-54. 
Suggestion.  —(9  -  x^y  =(x^  -  9)2. 

49.  (4-cy  andc3  +  9c2  +  26c  +  24. 

50.  (x  -  x%  (x'  -  If,  and  (1  -  xf. 

51.  (1  - y^y  and  (2/  +  Ifil  - y)\f  -Ty-^6). 

52.  xy  -  /,  -  (2/^  -  x-y),  and  a;^?/  -  a;/. 

53.  16-s^  2s-s2,  ands2-4s4-4. 
64.  2/*  —  ic*>  ^-h  y^,  and  2/^  +  2  2/3/'  +  a^. 

55.  ar^-(2/4-2:)^  (2/-a;)^-2;^  and2/2-(ie-2!)2. 

56.  (2/  —  a;)^(ri  —  m)^  and  (x-y  —  y^)  (m^n  —  2  mn^  +  ti^). 

When  some  of  the  given  expressions  are  difficult  to  factor, 
their  factors  may  often  be  discovered  by  dividing  by  those  of 
the  more  easily  factored  expressions. 

57.  (m  H-  2)  (m^  -  9)  and  m*  -  3  m^  +  3  mhi  +  3  m V  -  9  mhi  + 
mn^  —  9  mn^  —  3  n^. 

58.  6  x^ -Sx -4:5,  9 x^- S3 x-\-lS,Sind6af- 3 x'-Sdx-lS. 

59.  2a;^-a^-a^,  2a.-24-a;-3,  anda^-a^-a;  +  l. 

60.  a;*-4ar^4-2a^+«  +  6,  2a^-9  a.-2H-7a;+6,  Sindx^-5x+6. 

61.  s3-8,  ,93  +  s2^2s-4,  ands*  +  2s3_s2-10s_20. 

62.  ar^  +  f  and  a^  -  2  x'^y  -\-2x^y'^-2  xy  -\-2xy^-  f. 


LOWEST  COMMON   MULTIPLE 


180.  An  expression  that  exactly  contains  each  of  two  or 
more  given  expressions  is  called  a  common  multiple  of  them. 

0  abz  is  a  common  multiple  of  a,  3  6,  2  x,  and  6  abx.  These  numbers 
may  have  other  common  multiples,  as  12a6a;,  dd^b'h:^  ISa^bx^,  etc. 

181 .  The  expression  having  the  smallest  numerical  coefficient 
and  of  loivest  degree  that  will  exactly  contain  each  of  two  or 
more  given  expressions  is  called  their  lowest  common  multiple. 

Qabx  is  the  lowest  common  multiple  (L.C.  M.)  of  a,  3  6,  2ac,  and  Qabx. 
The  lowest  common  multiple  of  a  and  b  is  ab,  regardless  of  the  values  that 
a  and  b  may  represent.  What  the  arithmetical  least  couunon  multiple  is 
depends  upon  tlie  values  of  a  and  b.  If  a  =  6  and  6  =  2,  the  least  common 
multiple  is  not  12,  the  value  of  ab,  but  6. 

The  lowest  common  multiple  may  have  either  sign. 

In  §§  180,  181,  only  rational  integral  expressions  are  included. 

182.  Principle.  —  The  loivest  common  multiple  of  two  or  more 
expressions  is  the  product  of  all  their  different  prime  factors^  each 
factor  being  used  the  greatest  number  of  times  it  occurs  in  any  of 

the  ex})ressions. 

EXERCISES 

183.  1.   What  is  the  L.C. M.  of  12x^yii^,  6a*a^,  and  Saxyz*? 

Solution.  — The  lowest  common  multiple  of  the  numerical  coeflBcients 
is  found  as  in  arithmetic.     It  is  24. 

The  literal  factors  of  the  lowest  common  multiple  are  each  letter  with 
the  highest  exponent  it  has  in  any  of  the  given  expressions  (Prin.).  They 
are,  therefore,  a*,  x^,  y'^,  and  z*. 

The  product  of  the  numerical  and  literal  factors,  24a*a!'^yV,  is  the 
lowest  common  multiple  of  the  given  expressions. 

123 


124  LOWEST  COMMON  MULTIPLE 

2.   What  is  the  L.C.M.  oi  x^  -2  xy  -\-y\  y'  -x^^djudi  o^  -\-f? 

PROCESS 

o?-2xy  +  y^  ={x-y){x-y) 

2/2  _  a^  =  -  (a;2  _  2/2)  =  _  (cc  +  2/)  (^  -  y) 

^+f  ^(x  +  yXx'-xy-i-y^ 

L.  C.  M.  =(x-y)\x-{-y)(x'-xy  +  f) 

Rule.  —  Factor  the  expressions  into  their  pH me  factors. 

Find  the  product  of  all  their  different  prime  factors,  using  each 
factor  the  greatest  number  of  times  it  occurs  in  any  of  the  given 
expressions. 

The  factore  of  the  L.  C.  M.  may  often  be  selected  without  separating 
the  expressions  into  their  prime  factors. 

Find  the  lowest  common  multiple  of : 

3.  a^a^y,  a^xy^,  and  ax^y. 

4.  10  a^b'c^  5  ab%  and  25  b^c^d^. 

5.  16  a^b%  24  c^de,  and  36  a^bH^e^. 

6.  18  a26r2,  12^2^2,.^  ^nd  54  abYq. 

7.  x'^y^,  ic'"-y,  x"*-'Y,  and  x'^+^y. 

8.  x^  —  y^  and  x^-\-2  xy  -{-  y\ 

9.  x^  —  y^  and  x^ —  2  xy -\-y'^. 

10.  y?  —  y^j  01? -\-2  xy-\- y^,  and  ^  —  2xy -\-y'^. 

11.  a2  -  ^2  and  3  a^  +  6  a27i  +  3  a7i2. 

12.  cc^-l  anda2x2  +  a2-62a^_52^ 

13.  G?  +  1,  ab  —  b,  a*  4-  a,  and  1  —  al 

14.  2  .T  +  2/,  2  a;,?/  —  2/^  and  4  ic2  _  y2^ 

15.  1  -f-a;,  a;  —  ar^,  1  +  ar^,  and  a;2(l  —  a;). 


LOWEST   COMMON   MULTIPLE  125 

16.  2x4-2,  6a;  — 5,  3a;  — 3,  and  ic'  —  l. 

17.  166«-1,  12  62-1-35,  206-6,  and  26. 

18.  l-2a;*  +  a;^  (1-x/,  andl-f  2x  +  aj». 

19.  X2/-1/2,  ar'-ha:y,  xyH-y*,  anda;*  +  y*. 

20.  ^^  —  7?,  x^  +  xy  -\-  y*,  and  m?  —  xy, 

21.  6^-66  +  6,  6'-76  +  10,  and  6^-106  +  16. 

22.  :x^->rl  x-%yX^-l,x-{-Q^,2Liidi3aa?-(Sax-{-Z(L 

23.  ar'-a^  a-2x,  a2  +  2aa;,  and  a^-3a*x  +  2aa;2. 

24.  m^  —  X*,  iin?  +  mx,  m*  4-  wix  +  x*,  and  (m  +  x)ar*. 

25.  2-3x-|-x*,x2-f  4x44,  x*  +  3x  +  2,  and  1-x*. 

26.  x2_y2,  a;^4.a^y2  +  y4^a;»4.ys^andx2-|-a^4y. 

27.  x*  4  x*?/  4  xy*  4  y*  and  x*  —  a?y  4  xy-  —  y. 

28.  a-44a  +  4,  a2-4,  4-a2,  anda*-16. 

29.  a«  -  (6  4  c)\  6*  -  (c  4  a)^  and  c*  -  (a  4  6)^. 

30.  m  —  Tij  (m*  —  »^*,  and  (?w  4  n)^ 

31.  a"  -  6^  and  a»  4  a*6*H- 6*. 

32.  x«4/andaV-6y4ay-&V. 

33.  a*  -  a^  4 1,  a*  4  1,  a*  4  a-  4  1,  and  a'  —  1. 

34.  2(ax2  -  x^*,  3  x(a*x  -  x»)»,  and  6(aV  -  a*). 

35.  (y2*  —  xyz)',  y*(xz*  —  x^,  and  x*2*  4  2  xsr*  4  z*. 
Suggestion.  —  In  solving  the  following,  use  the  factor  theorem. 

36.  x>-6x*4lla;-6andx3-9x2426x-24. 

37.  x»-5x»-4x420andx'*42x»-25x-50. 

38.  x»-4x«45x-2andx*-8x*421x-18. 

39.  x''  4  5  X-  4  7  X  4  3  and  x'  —  7  x*  —  5  x  4  75. 

40.  x»4-2x»-4x-8,  x^-x2-8x4l2,  x^44x2-3x-18. 


FRACTIONS 


184.  A  fraction  is  expressed  by  two  numbers,  one  called  the 
numerator,  written  above  a  line,  and  the  other  the  denominator, 
written  below  the  line. 

If  a  and  b  represent  positive  integers,  as  3  and  4,  the  fraction 

-  is  equal  to  - ;  that  is,  it  represents  3  of  the  4  equal  parts  of 

anything.     This  is  the  arithmetical  notion  of  a  fraction. 

But,  since  a  and  b  may  be  any  numbers,  positive  or  negative, 

integral  or  fractional,  -  may  represent  an  expression  like  — . 

Since  a  thing  cannot  be  divided  into  5|  equal  parts,  algebraic 
fractions  are  not  described  accurately  by  the  definition  com- 
monly given  in  arithmetic.  But,  since  an  expression  like  -^^-, 
regarded  as  20  fourths,  is  equivalent  to  5,  or  20-4-4,  it  is  evi- 
dent that  the  numerator  of  a  fraction  may  be  regarded  as  a 
dividend,  and  the  denominator  as  its  divisor;  and  this  inter- 
pretation of  a  fraction  is  broad  enough  to  include  the  fraction 

-  when  a  and  6  represent  any  numbers  whatever.     Hence, 

The  expression  of  an  unexecuted  division,  in  which  the  dividend 

is  the  numerator  and  the  divisor  the  denominator^  is  an  algebraic 

fraction. 

The  fraction  -  is  read,  '  a  divided  by  &.' 

b  , 

185.  The  numerator  and  denominator  of  a  fraction  are  called 
its  terms. 

186.  An  expression,  some  of  whose  terms  are  integral  and 
some  fractional,  is  called  a  mixed  number,  or  a  mixed  expression. 

a  -  ^~  ^\  —  —  2  +  — ,  and  a  —  b  +  —  are  mixed  expressions. 
c        d^  x^  ab 

126 


FRACTIONS  127 

Signs  in  Fractions 

187.  The  sign  written  before  the  dividing  line  of  a  fraxition 
is  called  the  sign  of  the  fraction. 

It  belongs  to  the  f iiiotion  as  a  wholCy  and  not  to  the  numera- 
1(1  (u  to  the  denominator  alone. 

In  —  —  the  sign  of  the  fraction  is  — ,  while  the  signs  of  x  and  3  z  are  + . 

188.  An  expression  like  ^^  indicates  a  process  in  division, 

—  h 
in  which  the  quotient  is  to  be  found  by  dividing  a  by  6  and 
prefixing  the  sign  according  to  the  law  of  signs  in  division ; 
that  is,  -a_,a  ±^__u« 

—  a_     a  H-«_     «. 

~+b~      b*  -b~     b' 

By  observing  the  above  fractions  and  their  values  the  fol- 
lowing principles  may  be  deduced  : 

189.  Principles.  —  1.  77ie  signs  of  both  the  numerator  and 
the  denominator  of  a  fraction  may  be  changed  without  changing 
the  sign  of  the  fraction. 

2.  The  sign  of  either  the  num£rator  or  the  denominator  of  a 
fraction  may  be  changed^  provided  the  sign  of  the  frcwtion  is 
cJianged. 

When  either  the  numerator  or  the  denominator  is  a  polynomial,  ita  sign 
is  changed  by  changing  the  sign  of  each  of  its  terms.  Thus,  the  sign  of 
a  —  6  is  changed  by  writing  it  —  a  -h  b,  or  b  —  a. 

BXERCISBS 

190.  Reduce  to  fractions  having  positive  numbers  in  both 

terms : 

_3  -a-x       ^         -n-b        _  -2-m 

1.     •  6.     •         6. -— •  /. — • 

-4  2x  c  +  d  2  +  n 

o       2  ,        -4c  g  -2  g     -Ma  +  b) 


-b-y  -a-y  5(-a;-.y) 


128  FRACTIONS 

191.  By  the  law  of  signs  for  multiplication,  the  product  of 
two  negative  factors  is  positive ;  of  three  negative  factors,  nega- 
tive; otfour  negative  fsictors,  positive ;  and  so  on.     Hence, 

Principles.  —  3.  TJie  sign  of  either  term  of  a  fraction  is 
changed  by  changing  the  signs  of  an  odd  number  of  its  factors. 

4.  TJie  sign  of  either  term  of  a  fraction  is  not  changed  by 
changing  the  signs  of  an  even  number  of  its  factors. 

EXERCISES 

192.  1.   Show  that  (" -b)(d-c)^{a- b)(c ~ d)_ 

(o  —  a){b—  c)      (a  —  c)(6  —  c) 

Solution  or  Proof 

Changing  (d  —  c)  to  (c  —  d)  changes  the  sign  of  one  factor  of  the 
numerator  and  therefore  changes  the  sign  of  the  numerator  (Prin.  3). 

Similarly,  changing  (c  —  a)  to  (a  —  c)  changes  the  sign  of  the  denomi- 
nator (Prin.  3). 

We  have  changed  the  signs  of  both  terms  of  the  fraction.  Therefore, 
the  sign  of  the  fraction  is  not  affected  (Prin.  1). 

2.  Show  that  (6-«)('^-c)  =  _  («-6)(c-d), 

(c  —  b)(a  —  c)  (5  — c)(a  — c) 

Solution  or  Proof 

Changing  the  signs  of  two  factors  of  the  numerator  does  not  change  the 
sign  of  the  numerator  (Prin.  4). 

Changing  the  sign  of  07ie  factor  of  the  denominator  changes  the  sign 
of  the  denominator  (Prin.  3). 

Since  we  have  changed  the  sign  of  only  one  term  of  the  fraction,  we 
must  change  the  sign  of  the  fraction  (Prin.  2). 

3.  Show  that  -^ —  may  be  properly  changed  to 


b  —  a  a— b 


4.  From  derive  by  proper  steps. 

6  —  a-f-c  a  —  b  —  c 

5.  Prove  that —^  = ^:  that  -      ^  ^ 


x-1'  A-x'     aj2-4 


6.    Prove  that 


7.   Prove  that 


8.   Prove  that 


9.    I'rove  that 


FRACTIONS  129 

2  2 


a(h^a)         a(a  —  b) 

5x  5x 


{x  +  y)(2/  -  x)  (x  -f  y)(;x  -  y) 

2a    ^  2a 

9_a*     (aH-3)(a-3)* 

(5-«)(c_6)      (a-6)(6-c)* 


10.   Prove  that  (^-^)(^  +  ^)  =     -m^  +  n^    . 
(a  — c)(6— a)       (a  — c)(a— 6) 

11    Prove  that     ici-b){b-a  +  c)    ^    (a-b){a-h^c)    ^ 
iy-x){z-y){z-x)      (x-y){y-z)(x-z) 


REDUCTION   OF   FRACTIONS 

193.  The  student  will  find  no  difficulty  with  algebraic  frac- 
tions, if  he  will  bear  in  mind  that  they  are  essentially  the  same 
as  the  fractions  he  has  met  in  arithmetic.  He  will  have  occa- 
sion to  change  fractions  to  higher  or  lower  terms ;  to  write  in- 
tegral and  mixed  expressions  in  fractional  form;  to  change 
tractions  to  integers  or  mixed  numbers;  to  add,  subtract, 
multiply,  and  divide  with  algebraic  fractions  just  as  he  has 
learned  to  do  with  arithmetical  fractions. 

194.  The  process  of  changing  the  form  of  an  expression 
without  changing  its  value  is  called  reduction. 

195.  Principle.  —  Multiphjing  or  dividing  both  terms  of  a 

fraction  by  the  same  number  does  not  change  the  value  of  the 

fraction  ;  that  is, 

a  _  am       (mt  _  a 

b     bm        bin     b 

196.  A  fraction  is  in  its  lowest  terms  when  its  terms  are 
prime  to  each  other. 

M1LNK*S   8TAND.   ALO.  —  » 


130  FRACTIONS 

197.    To  reduce  fractions  to  higher  or  lower  terms. 

EXERCISES 

1.   Reduce to  a  fraction  whose  denominator  is  a^  —  h^. 

a  +  h 

PROCESS 

(a^-h'')^{a  +  h)  =  a-h. 

Then  g     ^       a{a-h)      ^a'-ab 

a  +  6      (a-\-b){a-b)      a'-b' 

Explanation.  —  Since  the  required  denominator  is  (a  —  b)  times  the 
given  denominator,  in  order  that  the  value  of  the  fraction  shall  not  be 
changed  (§  195)  both  terms  of  the  fraction  must  be  multiplied  by  (a  —  6). 

5a 

2.  Eeduce  —  to  a  fraction  whose  denominator  is  42. 

6 

Sx 

3.  Reduce  — —  to  a  fraction  whose  denominator  is  55  b. 

11  b 


8a 


4.   Reduce  to  a  fraction  whose  denominator  is  84 


14  a; 


xy. 


4a^ 

5.  Reduce  to  a  fraction  whose  denominator  is  20  y^. 

5y 

6.  Reduce to  a  fraction  whose  denominator  is  (x  —  lY. 

x  —  1 

7.  Reduce toafraction  whose  denominator  is  (2x-{-5y. 

2x-{-5 

8.  Reduce  -— —  to  a  fraction  whose  numerator  is  3  a  +  a^. 

o  —  a 

9.  Reduce ^  to  a  fraction  whose  numerator  is  x^  —  if. 

10.  Reduce  ^ —  to  a  fraction  whose  denominator  is  4  —  a^. 

x  —  2 

11.  Reduce  a;  —  5  to  a  fraction  whose  denominator  is  a;  -|-  5. 


FRACTIONS 


181 


21  a'a^v 

12.    Reduce — ^  to  its  lowest  terms. 

30  a»a» 


PROCESS 

I'l  a^x^y  _  7  xy 
30  a^xz  ~  10  az 


Explanation.  — Since  a  fraction  is  in  its  lowest 
terms  when  its  terms  are  prime  to  each  other,  the 
given  fraction  may  be  reduced  to  its  lowest  terms  by 
removing  in  succession  all  common  factors  of  its 
numerator  and  denominator  (§  195),  as  3,  a,  a,  and 
x;  or  by  dividing  the  terms  by  their  highest  common 
factor,  3  a^. 


Reduce  to  lowest  terms : 
16  vi^nx^z^ 


13. 


14. 


15. 


16. 


17. 


40  amhfT^ 

750  ah^c 

35  d'hcxJ^ 
42  aU'cd* 

11  d'x'l^y 
121  aW* 

-25a^y'z^ 


18. 


19. 


20. 


21. 


28. 


29. 


30. 


31. 


32. 


33. 


100  a?*/ 

a*  +  2a6  +  6*' 
a«_2a6+6* 


22. 


a' 
4  a*- 


-6« 


8a»  +  27a:» 
3a'H-3a6 

Sa^-6xy 
aih/  —  Sxy 

3  a*6  -  3  y 
2a»6-2  6*' 


a^xy 


arrrrr 

oW 
b'xf' 

^y' 


ar-^a' 


34. 
35. 
36. 
37. 
38. 
39. 


23. 


24. 


25. 


26. 


27. 


xy^^' 


X 


.m-M+I 


ax 

a'h" 
3a^b 


"y^ 


2a''y^ 


oaf* 


-w+l 


2x'u' 


6af— 
8y^ 


4  a^y  -  32  y 

10  na;  -V 10  ny 
25ns^-25nf' 

af-*-3— af* 
a'^+*  —  a^y^  ^ 

a^.y  —  g'y^  -h  y* 
a^  +  / 


132  FRACTIONS 


4^    a^-lla  +  24  ^^      a' -3a'b+Sab'-b' 

a'' -a -6     '  '  Sab''-3a^b 


9 


ar^  +  2x^-35ic  9  o^- 12  aa;4- 4  a^ 

^2     7  g;  —  2  a;^  —  3  2  qx  -  ay  —  4  6a.-  +  2  6y 

'^  a?-'  +  7  a?  —  4  '    4:ax  —  2  ay —  2  bx-\-by 


^g     a(a  +  2  ?>)V  gg         9a^-13a2a;-4a3 


b{a:'-4:¥f  3bx-\-3xy—4ab~4:ay 

. .     a^  +  2  a^6  +  a6^  ^  .  m  —  m^  —  7i  -}-  mn 

44.    -^— ' ■ •  54.  ^ • 

a^  —  2  a%^  -\-  ab^  m  —  mn  -\-n^  —  n 

.^     x^  —  2x'^-\-x^  ^^  am  —  an  —  m-\-n 

*^-     li i °°- 

x'  —  xr  am  —  an-\-  m  —  n 

.-     Qi?  -\-^x^  —  ^x  ^-  ar'  +  oa^  —  9a;  —  45 

4d.     ■ — r •  5d. 


2a:2_2  a^  +  3a^ -25a;-75 

^^      a^_7a;  +  6  ^  ^^       a^-f2  a^-23  a;- 60 


x^-llx'-10x  +  2m 


^^     20-21a;  +  a.'3  ^„  ^a^-lx'  +  l 

4o.     -— — - — -— •  5o. 


26  a^  +  25  5  a.-^  - 17  .T^  +  1 6  a;  -  4 

^g     ar^  +  3a^  +  3a;4-l  59     x^ -i^x'y +  2xy''  +  3  y^ 

4  4-  4  aj  —  X-  —  a.-^  *     5^/^  +  2  xy^—(5  Qi?y—Qi? 


60. 


61. 


62. 


63. 


a2  4- 52  _^  2  c^  4. 2  a6  +  3  ac  + 3  6c 


a2  +  6'  +  < 

32  +  2  a6  +  2  ac 

■^2  be 

a2  +  62_^c' 

2  +  2  (x6  -  2  ac  • 

-2  be 

a2  +  52_^2_^2a6 

g^  +  6^  +  c^  -  2  ad  -  2  ac  +  2  6c 
a2  +  62  +  5  c^  -  2  a6  -  6  ac  +  6  6c ' 

4  a2  4-  9  6^  + 16  c^  + 12  a6  + 16  ac  +  24  6c 
4a--96  +16c2  4-16ac 


FRACTIONS  138 

198.   To  reduce  a  fraction  to  an  integral  or  a  mixed  expression. 

EXERCISES 

1.  Reduce  ^'^  "'"    to  a  mixed  number. 

X 

PROCESS  Explanation. — Since  a  fraction  may  be  regarded 

,               ,  as  an  expression  of  unexecuted  division,  by  perfonn- 

JL_  =  a-f-_  ing  the  division  indicated  the  fraction  is  changed 

^                  ^  into  the  form  of  a  mixed  number. 

2.  Reduce  ^  ""'  ^  J"^        to  a  mixed  number. 

The  division  should  be  continued  until  the  remainder  is  lower  in  degree 
than  the  denominator  or  no  longer  contains  the  denominator. 

Reduce  to  an  integral  or  a  mixed  expression : 

2x 
.     ab  —  bc  —  cd  +  d^ 


b 


12. 


_     rt  V  —  ooj*  —  a;  —  1  - « 

O.      •  AO. 


ax 
a^-x-15 


14. 


x  +  2 

a»4-9a2-h24a-l-22 

aH-3 

3^-\-2a?-x'-\-5 

a^  +  x" 

4a^  +  12a»-a2-f-34 

aj-4  2a^-\-5 

7.     = '—  10.     — • 

x—y  a—Zb 

g     3^  —  6x3/4-43^  jg    x^  —  7x  — 4a;4-40 

2xy          '  '               a^-3 

g     si^-6a^-hUx-9  a'^  +  Sa^b-ab^  +  ab 

x-2            '  '               a^-\-b 

10     a^-3x'-f  5a;-l  ^g    a^-xi/-3f-z 

x  —  3  '            x-\-y 


134  FRACTIONS 

19     «'  +  3a^6^  +  &^  2j     x*-\-Aa^y-\-6x'y^-\-4:xf 

a^-^b^  '  x  +  y' 

2^     4a^  +  22a;  +  21  ^^     rn' -27n^n-Smn^-2mn 

2x  +  4:  '  m^  —  mn 

199.  To  reduce  dissimilar  fractions  to  similar  fractions. 

200.  Fractions  that  have  the  same  denominator  are  called 
similar  fractions. 

201.  Fractions  that  have  different  denominators  are  called 
dissimilar  fractions. 

202.  Principle.  —  The  lowest  common  denominator  of  two 
or  more  fractions  is  the  loicest  common  multiple  of  their 
denominators.  ■ 

The  abbreviation  L.  CD.  is  used  instead  of  lowest  common  denominator. 


203.     1.   Eeduce  -^  and  — ^  to  fractions  having  their  low- 
3  6c  6  ab 

est  common  denominator. 

PROCESS  Explanation.  —  Since  the  L.  C.  D.  of 

o  y  2  o  2  a'^      *^®  given  fractions  is  the  lowest  common 

777- =  77-7 7, —  =  7; — 7~      multiple  of  their  denominators  (Prin.), 

3  6c     3  6c  X  2  a     Qabc     ^,     ,        ^  ,^.  ,      ^»   ,,   . 

the  lowest   common    multiple   of    their 

c  cX  c  (?  denominators  must  be  found.     This  is 

e^'^ea^x  c^ ^abc       ^«^^- 

To  reduce  the  fractions  to  equivalent 
fractions  having  the  common  denominator  6  abc^  the  terms  of  each  fraction 
(§  195)  must  be  multiplied  by  the  quotient  of  6  abc  divided  by  the  denomi- 
nator of  that  fraction. 

Rule.  —  Find  the  lowest  common  multiple  of  the  denominators 
of  the  fractions  for  the  lowest  common  denominator. 

Divide  this  denominator  by  the  denominator  of  the  first  fraction, 
and  multiply  the  terms  of  the  fraction  by  the  quotient. 

Proceed  in  a  similar  manner  ivith  each  of  the  other  frax^tions. 

All  fractions  should  first  be  reduced  to  lowest  terms. 


FRACTIONS  185 

2.   Reduce  2  m  and  ^L±i^  to  fractions  having  their  L.  C.  D. 
m  —  n 

SuooBBTioN.  —  Firet  write  2  m  as  a  fraction  with  the  denominator  1. 
Reduce  to  similar  fractions  having  their  L.  C.  D. : 


3. 

|a„d^'. 

7. 

wt  —  n     ()         a 

4 

1^  and  3a;. 
56 

8. 

a?            X           X 

a53_l'    aj+i'    a;-l 

5. 

9. 

a»             a            2a 

a*-16'    a*  +  4'    ^-a' 

3       -6 

3 

4a         36           1 

6. 

xy'  :^2^' 

^/ 

10. 

a_6'    h+a     a'-b' 

11 

1 

1 

1 

•   ar'  +  Tar-hlO' 

a*  +  a;- 

-2'    ic»  +  4a;-5 

19          « 

^  +  5 

a-2                a-hl 

a*-4a  +  3'    a^-Sa  +  lS'   a*-6o  +  5 


ADDITION   AND    SUBTRACTION   OF   FRACTIONS 

204.  It  lias  been  learned  in  arithmetic  that  only  similar  frauo 
tions  may  be  tmited  into  one  fraction  by  addition  or  subtraction. 

The  method  of  adding  and  subtracting  similar  fractions  is 
much  the  same  in  algebra  as  in  arithmetic.  In  algebra,  how- 
ever, subtraction  of  fractions  practically  reduces  to  addition  of 
fractions,  for  every  fraction  to  be  subtracted  is  really  added 
with  its  sign  changed  (§  56,  Prin.). 

The  usual  method  of  changing  the  sign  of  a  fraction,  in  such 
cases,  is  to  change  the  sign  of  its  numerator  (§  189,  Prin.  2). 

Thus,  g  +  ^_£+-^°+^-''+-.    That  is, 

XXX  X 

205.  Principle.  —  If  fractions  have  a  common  denominator, 
their  sum  is  the  sum  of  their  numerators  divided  by  the  common 
denominator. 


136  FRACTIONS 


EXERCISES 

206.    1.    Add  ^,    ^,  and  ^l. 

PROCESS 

3  a;     7  a;     5y_45a;     42  a;     25  y 
4       10      12  ~  60        60        60 

60 
PvXPLANATiON.  —  SincG  the  fractions  are  dissimilar,  they  must  be  made 
similar  before  they  can  be  united  into  one  term.     The  L.  C.  D.  =  60. 

Then,  Sx^46x      7j  ^  42^    ^^^  5  y  ^  25j/_ 

4         60        10        60  12       60 

The  sum  =  i^  +  ^  +  ?^^  =  45a;  +  42  x  +  25  2/  ^  87x  +  25y 
60         60         60  60  60 


2.    Subtract  ^^  from  ^-^^  +  ^. 

7  8  4 

Solution 

5a;-l      x_  x-2  ^  35  a;  -  7      14^  _  8  a; -16 

8  4         7  56  56  56 


35  a;- 

-  7  +  14x- 

-(8x- 

-16) 

56 

35  a;- 

-7+  14x- 

-8x4-16 

56 
^41x  +  9 
56 
Suggestion. — When  a  fraction  is  preceded  by  the  sign  — ,  it  is  well 
for  the  beginner  to  inclose  the  numerator  in  a  parenthesis,  if  it  is  a  poly- 
nomial, as  shown  above. 

Rule.  —  Reduce  the  fractions  to  similar  fractions  having  their 
lowest  common  denoyninator. 

Change  the  signs  of  all  the  terms  of  the  numerators  of  fractions 
preceded  by  the  sign  —,  then  find  the  sum  of  the  numerators,  and 
write  it  over  the  common  denominator. 

Reduce  the  resulting  fraction  to  its  lowest  terms,  if  necessary. 


3. 

2fand 
5 

3a; 
2 

4. 

^a„d 

66 
5  ' 

5. 

If- 

3a 
26* 

6. 

r'a„d-2. 
Ix           3x 

Simplify : 

FRACTIONS  137 

Add :  Subtract : 

7.  ^  from  1^. 
6  3 

8.  i^from-^. 
9  2 

9.  -2?  from  5. 
10.    ^i±ifrom^il^. 


11  2a;  +  l      a;-2      a;-3      5-a; 

3       "^     4  6  2     * 

12  a;-2      a;-4      2-3a;     2a;H-l 

6  9  4  12     ' 

13  a^-1      a;-2      4g-3      \-x 

3  18  27  6     * 

,^     2-6a;  ,  4a;-l      5ir-3      \-x 
14. 1 — . 


a;-i-3      x-2      x-\      a;-h3 
4  5  10  6     * 

,^     l-2a  ,  2a-l      2a-a«  +  l 
16.    ^— +  — J ^ 

3-}-x-x^     1-x  +  x^     l-2x-2a^ 
4  6  3 

5  rt*  _|_  ft2 

18.    Reduce      ,      ,, —  2  to  a  fraction, 
a*  — 6* 

Solution 

a-^  -  62       1  a2  _  6-2 

^5a2  +  fea-2a2  +  26g 
a*  -  6* 

a«-6« 


138  FRACTIONS 

Reduce  the  following  mixed  expressions  to  fractions : 

a^  —  ah 


19. 

-1- 

20. 

-i 

*>1 

a'-c' 

1     ^  r 

c 

22. 

1-x 

-4.x. 

23.  a  — 

24.  a- 


26.    a-\-x 


b 

a  —  b  —  c 

2 

x" 


26.    a^-ab-\-b 


a  —  x 


3  a  +  h 

Perform  the  additions  and  subtractions  indicated : 

27.  ^  +  *^^  37.    l  +  l  +  -2i^-2. 

ab  be  X  l  +  x 

28.  ^^±^  +  !i=i.  38.    2a-36-i^+9M 
a— 6     a+6  2a+36 

29.  ^ZL-^-ln^.  39.   3a_2a,-8«'-^^, 

be  ac  3a-\-2x 

30    ^  +  ^      g  —  & 
a—b      a+b 

31.    o^  +  ^z-^-tl-'. 

32         ^      ^JZ_z. 

ic-2     x  +  2' 

33.    '1^  +  ^-3  +  i 


40. 

112 

x-1      x+1      ^ 

41. 

1             1            2a 

a  +  6      a-6      a2-62 

42. 

a^x     a  —  x       4:  ax 
a  —  x     a-\-x     a-  —  x- 

43. 

a  + 1              a  —  1 

5       2  6  a^  +  a  +  l      a--a  +  l 

34.    a;  +  l  +  ^^-  44.    3x-h  —  -f2x-^^ 


ax      \  ax 


35.    m-'^'  +  ^Vn.  45.      «-6    ^ol+A^ « 


m  —  n  2  (a  +  6)     a- 

36    j^      aa;  -  6a;  +  a?> .    ,  ^g     a +  33         6  10 

x^  a-  —  9a  —  oa-\-3 


47.    -n--!!— ^  + 
a  — 2      a  ■ 

SuooESTioN.  — By  Prill.  1,  §  189, 


FRACTIONS  139 


a         a-2 


a-2     a-t-2     4-0^ 
3 


4  —  a-     a- 


48.  ?±i  +  -l-  +  JL^. 
tt-1      a-f  1      1-a' 

49.  ^JL±1+     2  3 


50. 


51. 


52. 


aJ8._4  a;-2      2-a; 

g(a4-x)  3ax-x''  ^  ^^ 

a  —  x  x  —  a 

1  1.1 


a8  +  8     8-a3     4-a« 

5(x-S)  2(a;4-2)  a;-l 

a^-a;-2     a:2^42._^3      Q_x__a^' 


63.   Simplify  ^±^-l+--l^—. 

Solution 
By  redacing  the  first  fraction  to  a  mixed  number, 

x^  +  z+l  _  1  .         2ae        =  i  +        2ag        - 1+        2a; 
X*  -  «  +  1  «2  +  X  +  1  x2  -  X  +  1  x«  +  X  4-  1 

_        2x  2x       ^  4x(xg  +  l). 

X^-X  +  l       X2  +  X+1        X*+X2+1 

Suggestion.  —  Frequently,  by  reducing  one  or  more  of  the  given  frac- 
tions to  mixed  numbers,  the  integers  cancel  each  other  and  the  numerators 
are  simplified. 

g^     a^  +  2ab-hy'      ^    .     2  ab 


55. 


56. 


57. 


a'+b" 

.   ,^ 

'■-b" 

a«  +  3a6+2&* 

a«- 

13  &« 

a^^3ab-4:b' 

a*- 

16  6« 

x+1 1 x-1 
x  —  1     x-\-l 

x  +  2 
x-2 

x-2 

x  +  2 

x-hS     x-3 

aj  +  4 

x-4 

X— 3     a;+3     x  — 4     x  +  4 


140  FRACTIONS 

\  a  2  ab         4:  ab^ 


58. 


b      a  +  6      a^-{-  b'^     a*-^b* 


Suggestion.  — Combine  the  first  two  fractions,  then  the  result  and  the 
third  fraction,  ilien  this  result  and  the  fourth  fraction. 


59. 


60. 


61. 


62. 


Sum  = 


63. 


64. 


65. 


66. 


67. 


a-\-b      a  —  b        4  a6         8  ab^ 
a-b     a  +  b      a^  +  U'     a*  +  6* 

1            1            2b           2W 
a-b      a-^b     a'-\-b'     a'  +  b' 

a -\- X     ci' -{- x^     a  —  x     a^  —  x^ 

4  a^a;  +  4  aa^ 

a  —  x  '  a^  —  x'      a  +  x      a^  +  ar^ 

a^-x' 

^+y              y+^ 

z-\-x 

(^y-z){z-x)      {x-z){x-y)      (y-x){z-y) 

Solution 

x-\-y         _^  y  +  z _j_  z-\-x 

y  -  z)(z  -x)       {z-  x){x  -  y)       {x  -  y){y  -  z) 

^  (x-^  -  y^)  +  (y'  -  z'-)  +  (z^  -  x2) 
ix-y)(y-z)iz-x) 


=  0. 

ix-yXy-^){^~^) 

^1^1^ 

(b- 

-c)(a-c)      (c-a)(a-b)      (b-a)(b-c) 

a  +  1          1          ^  +  1          1          C4-1 

(a- 

-b)(a-G)      (b-c){b-a)      (a-c){b-c) 

c\ib                       b^ca                       a^bc 

c^- 

-  a)(b  -  c)      (6  -  a)(b  -  c)      (a  -  6)(a  -  c) 

b  —  c                    G  —  a                    a  +  b 

(6- 

-aXa-c)      (b-c)(a-b)      (a-c)(b-c) 

c-\-a                     b-\-c                    a  +  b 

(a  —  b)(b  —  c)      {c  —  a)(b  —  a)      {c  —  b)(a  —  c) 


FRACTIONS  141 


MULTIPLICATION   OF   FRACTIONS 

207.  Fractions  aiu  multiplied  iu  algebra  just  as  they  are  in 
arithmetic. 

Thus  ?x^-'l^. 

inus,  4  ""  2-4x2 

In  general,  £  x  ^  =  ?^  •     That  is, 

0     a     bd 

Principle.  —  The  product  of  two  or  viore  fractions  is  equal  to 
the  product  of  their  numerators  divided  by  the  product  of  their 
denominators. 

EXERCISES 

208.  1.    Multiply  ?^  by  a^-25, 

x-\-o 

Solution 
x-5 


2.    Multiply  ^±1  by  1  +     ^ 


x  +  2  x-f-l 

Solution 

_x  +  3  ^^ 

^^    x  +  1 
_x  +  3 

x+r 

General  Suooestions.  — 1.  Any  integer  may  be  written  with  the  de- 
nominator 1. 

2.  After  finding  the  product  of  the  numerators  and  the  product  of  the 
denominators  the  resulting  fraction  may  be  reduced  to  lowest  terms,  in 
many  cases,  by  canceling  common  factors  from  numerator  and  denomi- 
nator. It  is,  however,  more  convenient  to  remove  the  common  factors 
before  performing  the  multiplications. 

3.  Generally,  mixed  numbers  should  be  reduced  to  fractions. 


142 

Multiply : 

3. 

Sab 
4.XIJ    ^ 

4. 

5xy  , 

Sax 

10/ 

5. 

^''  by 
10  c^    ^ 

3  6c 

a^ 

FKACTIONS 


8.    by 


a^ 


itn+l 


9.    ^  by  ^^— . 
10.    -^  by      ^ 


a4-6         a  —  6 

6     ii?^bv-i^-  11         ^^'       bv^^~^^^'. 

30^2/     ^       16m2  *    20 -8a;    *^         x'y 

^     2ax  .          1062                            l-6a;4-5a;2        2-x 
7.    by — .  12.    — ^ — by 

12  by    ^  x"  x^-'dx  +  2      ^  1-x 

Simplify  each  of  the  following : 

a-\-b        a'^  —  ab      a'^  —  V 

a^  —  x^       a-\-x       a^  —  ax  +  x^  ^ 
a^  +  01:^      a^  —  x^         {a -\-  x)- 

15.    4a-6^       2a       ^^4af^-/ 
'    2  x-\-y     4:a^  —  ab  4 

16     P  +  2     3x^-27  4 

•  x-S       2p'-S      px  +  Sp 

17,    P'-<t  y^    P-g   X      ^^ 
{p-qf      p^-\-pq      p'^-{-q^ 

18     a^  +  8      a^  +  2a  +  4 

•  a«_8''a^'-2a  +  4 

19.    g'  +  g V  4-  a;^  ^^  a; 

a*  —  a:ii?  a^  —  ax-\-:ii? 

20        ^'+4  g^^  +  g  +  1  . 

•  a^  +  a^  +  l      a-'  +  2g4-2 


FRACTIONS  148 


a^ -\- db  +  ac  +  be    a'—  ax-\-ay  —  xy    x^—  x{y  —  a)  — ay 
ax  —  ay  —  7? -\' xy    a^-\-ac  +  ax-\-cx     a*—a(y  —  b)—by 


•    a^-Sx'  +  ldx-U  '    ar»-6a:*+lla;-6 

a^-3g3- 23  ar^  + 75 a;-50       a;«- 10  ar +  29 a;-20 

DIVISION  OF  FRACTIONS 
209.   The  reciprocal  of  a  number  is  1  divided  by  the  number. 
The  reciprocal  of  6  is  -  ;  of  6,  i  ;  of  (a  +  ft),       ^ 


6  '         '6'       ^     •     "  a  +  6 

210.  Since  -  is  contained  d  times  in  1,  f  c  times  - )  is  con- 
1  ^  d  \  dj 

tained  -  of  d  times,  or  -  times,  in  1 ;  that  is, 
c  c 

1  _t-£  — ^. 

(/       c 

Principle.  —  T%€  reciprocal  of  a  fraction   is   the  fraction 
inveiied. 

211.  Since  1 -^-  =  -,  and  a  =  1  •  a, 

d     c 

it  follows  that  a-j--  =  -'aora  — 

d      c  c 

Principle.  —  Dividing  by  a  fraction  is  equivalent  to  multi- 
plyinfj  by  its  reciprocal. 


U4  FRACTIONS 


EXERCISES 


212.    Divide 


1.  Iby^.  •         3.    Iby  ^--^. 

-^    X  ct  +  b 

2.  1  by  -'  4.    Iby  ^^^. 
Write  the  reciprocal  of : 

o.    — •  7.    — •  y.    ° 

b  n  b  —  y 

6.    ^^  8.    J-.  10.    i-. 

j9  3  m  ao 

11.   Divide  ^^  by  ^i^- 

Solution 

^2— l"x-l      (a;  +  l).^a:,--47^iB-'|r-2"    x  +  l' 

Simplify : 

_ „     5  w?i      10  m-/i  -  Q  «^  —  &^  a-  +  b- 


6  to       3  aar^  a^  — 2a6  +  6-      d^—ab 

12  a*b  ,  4:ax  ^^     x^-\-y^  .  x^  +  xj/-\-y\ 
'     25  ac    '  24  c^  '    o^  —  y^  '        x  —  y 

14.    — =-a6a;.  20.    — -^ — - 

7  wi-^a?  —  mxr      iiv^x^  —  a;" 

16.  {a^^<l^.  22.    f5!^6.Vr^!xa6 

aH-6  6  \b 

17.  (4  a +  2)---- 23.   (a  +  c)-   — —--- ,, 

5  a  \l  +  a;      1— .1- 


24. 


FRACTIONS  146 


^^      a:«-6ic»4-llaJ-6  aj»-13arH-12 

ar'  +  2a-*-19a;-20     a-'' + 10  x^  +  29  a;  +  20 

SuooKSTioN. —  Reduce  the  dividend  to  a  fractiou. 

n.  (.v^«).(.«.,-i,> 

Complex  Fractions 

213.  A  fraction  one  or  botli  of  whose  terms  contains  a  frac- 
tiou is  called  a  complex  fraction. 

It  is  simply  an  expression  of  unexecuted  division. 

EXERCISES 

214.  1.    Simplify  tlie  expression — • 

'i  y 

s..,.,T.oN.  -  =  «^?  =  ?xJ^  =  «i^. 

■''■       b     y     b     X     bx 
'J 

MILSK  »    ^iA.NO.    ALO. 10 


146  FRACTIONS 


Simplify : 

x-\-y  2  +  — 


3 

ah 

x^-f 

ah^ 

s 

c 

6  +  ^ 
a 

™     3  m 

4. 

iC 

6. 


46 


6. 


,  86 


^ 

'*$■ 


^  10.       2/  ^' 


8 

'-I 

i- 

9, 

--f 

~-ax 

X  ^     .    1  11 

X 1+-  

m  a;  y     ^ 


11.    Simplify  the  expression 


^  +  -  +  1 


r    y 

Solution.  —  On  multiplying  the  numerator  and  denominator  of  the 
fraction  by  y^^  which  is  the  L.  C.  D.  of  the  fractional  parts  of  the  nu- 
merator and  denominator,  the  expression  becomes 


x^  —  xy  +  y^ 


x^  +  xy  +  y^ 


Sin 

iplify: 

12. 

X 

x^-1 

0^ 

13 

X     y-{-z 

1         1 

X     y-\-z 

o^  +  f 

x^  —  xy-{-  y^ 
xy 


15.    ^±1_.  17. 


:^  +  f 

22/ 

X     y 

y     X 

1 

1-a 

a 

a  +  1 


FRACTIONS  14' 

Simplify  : 

a;_24--^  Sa-l-- 

18 i+^ 


x-2 


14      4 

19.    ^      ^      ^. 
1  +  -  +  S 

X        XT 


20 

a 

2a-l 

3a 

21 

^-'-1 

9-3a; 

22.        a^-Hl  x  +  1      1-x 

^  _     1  X  a; 


l-fa     1— a;     14-a; 


23, 


3xyz 


x—1 , w— 1 , 2—1 


yz  +  zx-\-xy  1_^1_^1 

X      2/       2 


12  9  a^  +  (a4-6)a;  +  a6 

a;-|-?/      iK-y     3a;-2^  „_     x' -  (a -{- b)x  +  ab 

^^'  -^^87 ^^'  1^ 

2/2-9ar^  a^-a? 


26.    ^L_^±f 


^_^624-c2-a=' 


27, 


a     6  +  c  2  6c 

if 

{x  +  yY-:?     {x-y  +  zf' 
f  4 


a 


215.   A  complex  fraction  of  the  form  is  called  a 

continued  fraction.  h  4- 


d  + 


[S 

FRACTIONS 

28. 

Simplify  _ 

1 

X 

Solution 

1 

1 

1+     ' 

1  + 

1   '^i 

X 

1 


1  +    ^ 


x  +  1 

a;  4-  1  a;  +  1 


x+l  +  aJ      2x+l 

Suggestion.  — In  the  above  exercise,  the  part  first  simplified  is  the  last 
complex  part  ,  which  is  reduced  to  a  simple  fraction. 

X 

Every  continued  fraction  may  be  simplified  by  successively  reducing  its 
last  complex  part  to  a  simple  fraction. 

Simplify : 

29.     L__.  32.     '"-^ 


x  + x-2- 


1  +  |±1  ^_^ 

3  —  x  x  —  2 

30.     ^  33.  ^ 


a  +  14- ^  ""-^-^1 

a-\-l  —  -  a  +  i 

a  a 


31. 34.    1  + 


2 ^^^  l  +  c  +  ^ 


1  +  ^ 


2-a;  c 


REVIEW  149 

REVIEW 
216.    1.    When  is  a  fraction  in  its  lowest  terras  ? 

2.  Define  factoring ;  prime  factors ;  reciprocal  of  a  number. 

3.  When  may  the  factor  theorem  be  used  to  advantage  ? 

4.  Define  highest  common  factor ;  lowest  common  multiple. 

5.  Give  a  rule  for  finding  the  lowest  common  multiple  of 
two  or  more  expressions. 

6.  Show  that  the  product  of  a'  —  6'^  and  a^  —  h^  is  equal  to 
the  product  of  their  highest  common  factor  and  lowest  common 
multiple. 

7.  Distinguish  between  an  integral  and  a  fractional  alge- 
braic expression. 

8.  By  what  must  a  fraction  be  multiplied  in  order  to  obtain 
the  lowest  possible  integral  expression  ? 

9.  Why  must  a  broader  definition  of  fraction  be  given 
in  algebra  than  in  arithmetic?    Give  the  algebraic  definition. 

10.  Under  what  conditions  may  the  sign  of  the  numerator 
or  of  the  denominator  of  a  fraction  be  changed  ? 

11.  Show  that  r-^,=  -f^- 

12.  Show  that  the  sum  of  a  and  h  divided  by  the  sum  of 
their  reciprocals  equals  ah. 

13.  Distinguish  between  a  complex  fraction  and  a  continued 
fraction. 

14.  Show  that  dividing  _  by  x  is  the  same  as  multiplying 

r,   1  y 

-  by  -. 
y         X 

Simplify  : 

15.   ?_.  16.    ^.  17.    -    ^ 


XT-*  a»  -2»- 


150  REVIEW 

Reduce  to  lowest  terms : 


18. 


19. 


a^  +  Sx'  +  Bx  +  S 

a^  —  oi^  —  x  —  2 
a^  +  3x^-{-3x-\-2 


20. 


21. 


a^  +  a^-22a;-40 

a^-7x'  +  2x-i-^0' 

a^  +  10ar^  +  7a;-18 


Simplify : 


22. 


23. 


24. 


25. 


26. 


27. 


y 


2y-l'  2y  +  l      l-4.y 


4(l-a)2 

2 


-a)^8(a4-l)      4(a  +  l)y 


1  + 


8(1 

m^  +  m  —  2 


m  —  1  y  V    m^  +  m 
a-\-b      a  —  b        4:ab  \a^  —  b^ 


b      a  +  b      ct'+by    862 


2/y  V2/    y    ^. 


(x+i+i+i 


ic      a;^ 


x     x") 
\a  —  b     or  —  WJ  \a  —  b     a^  —  b^J 


29.    [x^-3xy-2y^-\- 


122/^ 

a^  +  32/. 


30.  r^-^^Vi+  ^^ 


m  +  n 


)0 


m4- w 


-(3a;-62/ 


15n 
m 


2a^ 


a;  +  '3?/ 


31     1      2a;  +  5a;2 
2(0^  +  1)^ 


ic^  +  aJ 


2a;- 


^  +  2 
m 


3-3x 


32.  fi+-^y^ 


X) 

2^-\-2ax- 
01?  -\-3ax-\- 


-af\ 
2aV 


REVIEW  151 

Expand  by  inspection : 

Simplify : 


37. 


i+i+i  «° 


ar^oj*  Ai     x-^y     x-\-y 


i+i+^  ' 


X     Q?  (a;  +  yf 


38.    ^ „   '    „^  ^^.  42.    1- 


?/r 

+  n« 

m^ 

-n* 

(a  +  iy 

^(^ 

!  +  !)» 

a 

^(^ 

1 

(o 

^4-1/ 

1 

1 

1 

1- 

1 

_ 

a^4-3aT^2 

a^  +  2a  +  l 
g'^  4-  7  g  H-  12 


39.    yti-h-^;      v^-t--^;  43. 


40. 


^^^^A  __2_n_\ 


2mn 


1  + 


1 


2— A 


\-x 
46 


/g-hy  ,  X - y\     /    x^y    _    x-y    \ 
\x-y     x  +  y)'\2{x-y)      2x-^2yJ 

\ar     x     a     a^J\7f     x     a? J 
x'-l  1  +  J- 

47.  ^2^,   ^^<v'    ^^rzl. 


SIMPLE   EQUATIONS 


ONE   UNKNOWN   NUMBER 

217.  The  student  has  already  learned  what  an  equation  is 
(§  4),  and  he  has  solved  many  equations  and  problems.  In 
this  chapter  and  the  next,  however,  he  will  find  a  more  com- 
plete and  comprehensive  treatment  of  the  subject,  extended  to 
some  kinds  of  equations  that  are  new  to  him. 

218.  An  equation  all  of  whose  known  numbers  are  expressed 
by  figures  is  called  a  numerical  equation. 

219.  An  equation  one  or  more  of  whose  known  numbers  is 
expressed  by  letters  is  called  a  literal  equation. 

220.  An  equation  that  does  not  involve  an  unknown  number 
in  any  denominator  is  called  an  integral  equation. 

a;  +  5  =  8  and h  5  =  8  are  integral  equations.     Though  the  second 

o 

equation  contains  a  fraction,  the  unknown  number  x  does  not  appear 
in  the  denominator. 

221.  An  equation  that  involves  an  unknown  number  in  any 
denominator  is  called  a  fractional  equation. 

8  2  r 

X  +  5  =  -  and =  7  are  fractional  equations. 

X  X  —  1 

222.  An  equation  whose  members  are  identical,  or  such  that 
they  may  be  reduced  to  the  same  form,  is  called  an  identical 
equation,  or  an  identity. 

a  +  h  =  a  -\-  h  and  a^  -  h"^  =  {a  -\-  b)(a  —  b)  are  identical  equations. 
An  equation  whose  members  are  nuuierical  is  evidently  an 
identical  equation. 

10  =  6  +  4  and  8x  2  =  6+ 12  —  2  are  identical  equations. 

152 


SIMPLE  EQUATIONS  153 

A  literal  equation  that  is  true  for  all  values  of  the  letters 
involved  is  an  identical  equation,  or  an  identity. 

(x  +  y)^  =  x^  -h  2  xy  +  y-  m  an  identity,  because  it  is  true  for  all  values 
iif  X  and  y. 

223.  An  equation  that  is  true  for  only  certain  values  of  its 
letters  is  called  an  equation  of  condition. 

An  equation  of  condition  is  usually  termed  simply  an  equa- 
tion. 

x  +  4  =  10  is  an  equation  of  condition,  because  it  is  true  only  when  the 
value  of  X  is  6.  x'^  =  9  is  an  equation  of  condition,  because  it  is  true  only 
when  the  value  of  x  is  -|-  3  or  —  3. 

224.  When  an  equation  is  reduced  to  an  identity  by  the 
ibstitution  of  certain  known  numbers  for  the  unknown  num- 
bers, the  equation  is  said  to  be  satisfied. 

When  a;  =  2,  the  equation  3  x  +  4  =  10  becomes  6  +  4  =  10,  an  identity; 
consequently,  the  equation  is  satisfied. 

225.  Any  number  that  satisfies  an  equation  is  called  a  root 
of  the  equation. 

2  is  a  root  of  the  equation  3  x  +  4  =  10. 

226.  Finding  the  roots  of  an  equation  is  called  solving  the 
equation. 

227.  An  integral  equation  that  involves  only  the  first  power 
of  one  unknown  number  in  any  term  when  the  similar  terms 
have  been  united  is  called  a  simple  equation,  or  an  equation  of 
the  first  degree. 

3  X  -h  4  =  10  and  x  +  2y  —  2;  =  8are  simple  equations. 

For  reasons  that  will  be  apparent  later  on,  simple  equations 
1  re  sometimes  called  linear  equations. 

228.  Two  equations  that  have  the  same  roots,  each  equation 
having  all  the  roots  of  the  other,  are  called  equivalent  equations. 

X  +  3  =  7  and  2  x  =  8  are  equivalent  equations,  each  being  satisfied  for 
X  =  4  and  for  no  other  value  of  x. 


154  SIMPLE   EQUATIONS 

229.  By  the  axioms  in  §  68,  if  the  members  of  an  equation 
are  equally  increased  or  diminished  or  are  multiplied  or 
divided  by  the  same  or  equal  numbers,  the  two  resulting  num- 
bers are  equal  and  form  an  equation.  But  it  does  not  neces- 
sarily follow  that  the  equation  so  formed  is  equivalent  to  the 
given  equation. 

For  example,  if  both  members  of  the  equation  a;  +  2  =  5,  whose 
only  root  is  x  =  3,  are  multiplied  by  x  —  1,  the  resulting  numbers, 
(x  +  2)(x  — 1)  and  5(x  —  1),  are  equal  and  form  an  equation, 

(x  +  2)(x-l)  =  5(x-l), 
which  is  not  .equivalent  to  the  given  equation,  since  it  is  satisfied  by 
X  =  1  as  well  as  by  x  =  3  ;  that  is,  the  root  x  =  1  has  been  introduced. 

In  applying  axioms  to  the  solution  of  equations  w^e  endeavor 
to  change  to  equivalent  equations,  each  simpler  than  the  pre- 
ceding, until  an  equation  is  obtained  having  the  unknown 
number  in  one  member  and  the  known  numbers  in  the  other. 

The  following  principles  serve  to  guard  the  student  against 
introducing  or  removing  roots  without  accounting  for  them : 

230.  Principles.  — 1.  If  the  same  expression  is  added  to  or 
subtracted  from  both  members  of  an  equation,  the  resulting  equa- 
tion is  equivalent  to  the  given  equation. 

2.  If  both  members  of  an  equation  are  multiplied  or  divided 
by  the  same  known  number,  except  zero,  the  resulting  equation  is 
equivalent  to  the  given  equation. 

3.  If  both  members  of  an  integral  equation  are  midtiplied  by 
the  same  unknown  integral  expression,  the  resulting  equation  has 
all  the  roots  of  the  given  equation  and  also  the  roots  of  the  equa- 
tion formed  by  placing  the  multiplier  equal  to  zero. 

It  follows  from  Prin.  3  that  it  is  not  alloioahle  to  remove  from  both 
members  of  an  equation  a  factor  that  involves  the  unknoinn  number, 
unless  the  factor  is  placed  equal  to  zero  and  the  root  of  this  equation 
preserved. 

Thus,  if  X  —  2  is  removed  from  both  members  of  the  equation 
(x  —  2)  (x  +  4)  =  7(x  —  2),  the  resulting  equation  x  +  4  =  7  has  only 
the  root  x  =  3  ;  consequently,  the  root  of  x  —  2  =  0,  removed  by  divid- 
ing by  the  factor  x—  2,  should  be  preserved, 


SIMPLE   EQUATIONS  165 

Clearing  Equations  of  Fractions 

231.  The  process  of  changing  an  equation  containing  frac- 
tions to  an  equation  without  fractions  is  called  clearing  the 
equation  of  fractions. 

EXERCISES 

Q  a* 

232.  1.   Solve  the  equation  — -—  =  6  —  -• 

Explanation.  —  Since  the  first  fraction 
PROCESS  ^jj  become  an  integer  if  the  members  of  the 

J.  _  g  jp  equation  are  multiplied  by  2  or  a  multiple  of 

— Ty —  ==  "  "~  o  2,  and  since  the  second  fraction  will  become 

an  integer  if  the  members  of  the  equation 
3aj  —  24  =  36  —  2a;  ^re  multiplied  by  3  or  a  multiple  of  3,  the 

5  a;  =  60  equation  may  be  cleared  of  fractions  in  a 

a;  =  12  single  operation  by  multiplying  both  mem- 

bers by  some  common  multiple  of  2  and  8, 
as  6,  or  12,  or  18,  etc.  It  is  usually  best  to  use  the  L.C.M.  of  the  denomi- 
nators, which  in  this  case  is  6. 

Then,  multiplying  both  members  by  6,  transposing  terms,  and  dividing 
by  the  coefficient  of  x,  we  obtain  aj  =  12. 

Verification.  —  When  12  is  substituted  for  x,  the  given  equation  be- 
comes 2  =  2,  an  identity  ;  consequently,  §  224,  the  equation  is  satisfied 
by  X  =  12. 

2.   Solve  the  equation  ^^  -  ^^  =  ?  -  ^'^  • 

Solution 
x~l     x-2_.2     x-3 
2  3        8         4     * 

Multijlyiii-:  both  members  of  the  equation  by  the  L.C.M.  of  the  de- 
nominators, which  in  this  case  is  12,  we  obtain 

tt  (X  -  1)  -  4  (x  -  2)  =  8  -  8  (X  -  3). 

Expanding,  6x-6 -4x-f-8  =  8  -  3x-|-9. 

Transposing,  etc.,  6x  =  15. 

Hence,  *  x  =  3. 

Verification.  —  When  x  =  3,  the  given  equation  becomes  f  =  f ,  an 
identity  ;  consequently,  the  equation  is  satisfied  by  x  =  3. 


156"  SIMPLE   EQUATIONS 

To  clear  equations  of  fractions : 

Rule.  —  Multij^ly  both  members  of  the  equation  by  the  lowest 
common  multiple  of  the  denominators. 

1.  To  simplify  the  work  and  to  avoid  introducing  roots,  reduce  all  frac- 
tions to  their  lowest  terms  and  unite  fractions  that  have  like  denomi- 
nators before  clearing. 

2.  If  a  fraction  is  negative,  the  signs  of  all  the  terms  of  the  numerator 
must  be  changed  when  the  denominator  is  removed. 

3.  Roots  are  sometimes  introduced  in  clearing  of  fractions.  Such 
roots  may  be  discovered  by  verification.  Those  which  do  not  satisfy 
the  given  equation  are  not  roots  of  it,  and  should  be  rejected 

Solve,  and  verify  each  result : 

3.  2x'  +  -  =  —  7.    ^  +  -  =  — 

3       3  2      6       3 

4.  -4-10-13.  8.    71-1^'  =  -. 
4  ^       14      7 

5.  -  +  2a^  =  26.  9.   ^-  +  ^=24. 

6  6     7 

6.  3x-^-  =  14.  10.    ^^-^  =  ^. 

5  3       6     4 

^^    X  ,  X      X  ,  Sx     5x      r- 
"•2  +  3-4+10-12  =  '- 

^.    25x     5x  .  2x     ox     c 

2x      7  g;      5  .^'       a;  _  4 
■  ^3        8"      TS      24""9* 

30^7  it'     X     9aj_l 
'  T      16~2'~16""8' 

\ox  I  6  0!  _  11  X     19 X _  2 

2x  ^x      4a;a;_l 
*   15      25      T      6~9* 


SIMPLE   EQUATIONS  167 

Sx      7  x^Ux      Sx      8 
4       12       36        9       2' 

ig    y-1  ,  y-2  ■  y-3_5y-l 

19.  ?i±l_!Hl4  +  !i±3^16. 
3  5  4 

72-4-2      12  —  2     2  +  2      ^ 

21  u-3     u-\-5     u  +  2_, 
'7^3  6 

22  3<-5     7<-13^»     <-}-3 

•  4  6  2    * 

23.  ^^-(a.-^n  =  3^  +  ^9_/^_^gY 

24.  1.07a:  +  .32  =  .15.TH-10.124-.675a;. 
SuooKSTioN.  — Clear  of  decimal  fractions  by  multiplying  by  1000. 

25.  .<;(4  X'  -  3.16  -  .7854  x  +  7.095  =  0. 

26.  3. 1416  X  -  15.5625  +  .021()  x  =  .2535. 

27  -2^     '^^     .la;  .  .4a;_  .3 

*  7         4         2  "^   7   ~14* 

28  n-f 4      2-2n^n4-l      10 

.3  .6  .2         .3* 

29  9a;-l-5     8a?~7^36g4- 15     10} 

14         6x+2  56  14* 


SuooESTioN.  — The  equation  may  be  written 

»jc  ,  _5   ,  8a;-7  _  S6x 

14      14     6x  +  2       56      56  "  50 


§130,  «^|    ^>   ,8x-7^36x,15  ,  41 


158  SIMPLE   EQUATIONS 

3a;-2     3  a;- 21  ^6  a;- 22 

'    2x-5  5  10      * 

4a;  +  3^8a;4-19      7a;-29 
9  18  5  a;- 12' 

6p  +  l       2p-4:  ^2p-l 
15         7p-13~      5 

lOg  +  17     5.g-2^12g-l 
18  9  11  g  -  8 

6r  +  3      3r-l  ^2r-9 
15         5r-25  5 


31. 


32. 


33. 


34. 


35.    Solve  the  equation  ^^  +  ^^  =  ^^  +  ^^  • 
^  a;-2     a;-7      a;-6      a;-3 

Solution.  —  It  will  be  observed  that  if  the  fractions  in  each  member 
were  connected  by  the  sign  — ,  and  if  the  terms  of  each  member  were 
united,  the  numerators  of  the  resulting  fractions  would  be  simple.  The 
fractions  can  be  made  to  meet  this  condition  by  transposing  one  fraction 
in  each  member. 

Consequently,  it  is  sometimes  expedient  to  defer  clearing  of  fractions. 

Transposing, 
Uniting  terms, 


X 

- 

1 

X  — 

■2 

_x_ 

5 

X  — 

6 

X 

- 

2 

X- 

3 

X 

- 

(3 

X  — 

7 

- 

-1 

— 

1 

a:2  -  5  X  H-  6     x'^  -  18  x  +  42 

Since  the  fractions  are  equal  and  their  numerators  are  equal,  their 
denominators  must  be  equal. 

Then,  x;^  -  5x  +  6  =  x^ -lSx-\- 42. 

.-.  X  =  41. 

_-     x  —  1  .  x  —  1      x  —  5  .  x  —  S 

Ob.    —  H -  — -i -• 

x  —  2      x  —  S      x—o      x  —  4: 

ic  —  3,cc— 7_ic  —  6,a;  —  4 

X  —  4:        X   -   Q        X—  i         X  —  D 


38. 


v-\-l      v-\-2     v4-4      v-\-o 


SIMPLE   EQUATIONS  159 


39     .'<  +  l   ,  s-i-G^s  +  2      8  +  5 
8-1-2     «-f7     «-h3     s  +  6 

X+1        X  —  1        X'—  I 

42.    |(2-r)-^(3-2r)  =  '"+^i5. 

.,     3n-4     /4n,  «  +  2\     9n     Aq  ,  n  +  4 


7  3  V  21     y      21 

16-')- 


44. 
^-         2  3      ^2 


JL_16     A  +  6 
24  60  5 

3         +         4         -    4     • 
48.    ^-2(^-3)=4-|(|+l). 

49     (2a;H-iy     (4a;-l)^^15      3(4x4-1) 
.05  .2  .08  .4 

17+5     1+1§     21_,      100^5 
50.    ^^4-^  =  ^^  +  ^       ^ 


15 

2a; 

4ra;-4^      4a;-16      3 
51. 


+  5 
^(a;-4)      4a;-16_3       o 


t  6  5  I 


160  SIMPLE    EQUA'IIONS 

Literal  Equations 

233.     1.    Solve  the  equation  ^^  =  ^^^  for  x. 

a  h 

Solution 


a  b 

Clearing  of  fractions,  bx—  b^  =  ax  —  a^. 

Transposing,  etc.,  ax—bx  =  a^  —  b^. 

(a  —  b)x  =  a^  —  ?)^. 
Dividing  by  (a  —  &),  x  =a^  +  ab  -\-  b^. 

Verification.  —  Whena  =  2and  b  =  l,x  =  aP--\-  ab  -{-b^  =4+2  +  1=7, 

7—1       7—4 
and  the  given  equation  becomes  — ^ —  = ,   or  3  =  3,  an    identity  ; 

consequently,  the  equation  is  satisfied  hy  x  =  a^  +  ab  -{-  b^. 

Solve  for  x,  and  verify  each  result : 

^     c'—  X  ,  n-      1  „     x  —  a  ,  2x      tr.66 

nx        ex      G  ha  a 

3  i_^  =  _I_i?..  8     ^'  [    y"  ^a-\-h      3(a+6) 

X       ah     ahx  hx     ax        ah  x 

4  _^_?_^'  =  i_?A'  9     ^'  +  &'     a-h^b 
ah^      h^x             a^x  2  hx        2  hx"^      x 

^     X    '  X  -\-2h      a      o  ,^     2x  —  a     x  —  a     ^ 

5. = O.  10. =  1. 

h  a  h  X  —  a       x-\-a 

X  —  2  ah      1  _x  —  3  c  x  —  2a     x:  __  a^  +  b^^ 

ex  x        ahx  '         a  h         ah 


12.  Q>x-{-l%(l--\  =  a{x-a). 

13.  h{2x-^e-li:h)=c{c-x), 

14.  a{x-a  —  2h)  +  h{x—l))-\-c{x^-e)  =  0. 


SIMPLE  EQUATIONS  161 

15.  ((I  -  x)(x  -  6)  -I-  (rt  -h  x)(^x  -  b)  =  (a  —  by. 

16.  I  (/  —  b){x  —  c)  —  {b  —  c)(x  —  a)  =  (c  —  a)(x  —  6). 

17. 


a  —  6  4-  c      b  —  a  -\-  c 


X  +  a  X  —  a 

18.    -J—  +  ^ 1 ^0. 

a(b  —  x)     b(c  —  x)     a(c  —  a;) 


19. 


20. 


X  —  1      a— 1  a^  —  a 


t/  —  1      j;  —  1      (a  —  1  )(x  —  1) 

1  2mn  m  x  —  n 


w  -\-n      (m  -h  ?i)^     (m  -f-  n)-      (m  +  n)* 

21.  a;-h«  I  g  +  c     a;4-6_ft  ■  ^  ■  c   ^ 

6  a  c         6      c      a       * 

22.    ^ + T =  a'  +  b'-\-c'-{-2ab, 

a-f  6  +  c     a-^b  —  c 

23    ^  -f  a;       2x        a?{x  —  a)  ^  1 
a -h  a;     a(a=*  -  3r2)      3* 

Sif;(;KSTi«)N   — Simplify  a.s  much  as  possible  before  clearing  of  frac- 
tions. 

x^-ax-bx  +  ah     a^-2bx-h2b-        ^ 


24. 


x  —  a  x—b  x  —  c 


Algebraic  Representation 
234.     1.    What  part  of  m  —  n  is  p  ? 

2.  From  what  number  must  m  be  subtracted  to  produce  n  ? 

3.  How  much  less  is  —  dollars  than  7n  dollars  ? 

n 

4.  Indicate  the  sum  of  I  and  m  divided  by  2,  and  that  result 
multiplied  by  n. 

5.  Indicate  the  product  of  s  and  (r  —  1)  divided  by  the  nth 
power  of  the  sum  of  t  and  v. 

6.  A  boy  who  had  m  marbles  lost  -  of  them.     How  many 
marbles  had  he  left  ?  " 

milneVs  stand,  alo.  —  11 


162  SIMPLE   EQUATIONS 

7.  By  what  number  must  x  be  multiplied  that  the  product 
shall  be  2  ? 

8.  Indicate  the  expression  for  -i-  the  product  of  g  and  the 
square  of  t. 

9.  Indicate  the  square  of  x,  plus  twice  the  product  of  x  and 
y,  plus  the  square  of  y,  divided  by  the  sum.  of  x  and  y. 

10.  By  what  number  must  the  sum  of  x  and  —  2/  be  multi- 
plied to  produce  the  square  of  x  minus  the  square  of  y? 

11.  Indicate  the  result  when  the  sum  of  a,  b,  and  —  c  is 
to  be  divided  by  the  square  of  the  sum  of  a  and  b. 

12.  It  is  t  miles  from  Albany  to  Utica.  The  Empire  State 
Express  runs  s  miles  an  hour.  How  long  does  it  take  this  train 
to  go  from  Albany  to  Utica  ? 

13.  A  cabinetmaker  worked  x  days  on  two  pieces  of  work. 
For  one  he  received  v  dollars,  and  for  the  other  iv  dollars. 
What  were  his  average  earnings  per  day  for  that  time  ? 

14.  A  train  runs  x  miles  an  hour  and  an  automobile  x  —  y 
miles  an  hour.  How  much  longer  will  it  take  the  automobile 
to  run  s  miles  than  the  train  ? 

15.  Indicate  the  result  when  b  is  added  to  the  numerator  and 

subtracted  from  the  denominator  of  the  fraction  -. 

c 

16.  A  farmer  had  -  of  his  crop  in  one  field,  -  in  a  second, 
^  x  y 

and  -  in  a  third.     What  part  of   his  crop  had   he  in  these 
three  fields  ? 

17.  A  won  m  more  games  of  tennis  than  B,  and  B  won  n 
more  games  than  C.  How  many  more  games  did  A  win 
than   C? 

18.  A  student  spends  —  of  his  income  for  room  rent,  -  for 

m  n 

board,   -  for  books,  and  -  for  clothing.     If  his   income   is   x 

s  r 

dollars,  how  much  has  he  left  ? 


SIMPLE  EQUATIONS  163 

Problems 

235.  He  view  the  general  directions  for  solving  problems 
:iven  in  §  77. 

1.  A  grocer  paid  $  8.50  for  a  molasses  pump  and  5  feet  of 
tubing.     He  paid  12  times  as  much  for  the  pump  as  for  each 

»ot  of  tubing.     How  much  did  the  pump  cost  ?  the  tubing? 
Suggestion.  —  If  we  knew  the  cost  of  a  foot  of  tubing,  we  could  com- 
I)ute  the  cost  of  the  pump.    Therefore,  let  x  represent  the  number  of  cents 
one  foot  of  tubing  cost. 

2.  A  merchant  purchased  an  assortment  of  bath  robes  for 
$  480.  By  selling  J  of  them  at  $  6  each,  ^  of  them  at  $  7  each, 
J  of  them  at  $  5  each,  and  \  of  them  at  $  8  each,  he  gained 
$  128.     How  many  bath  robes  did  he  sell  at  each  price  ? 

3.  A  shipment  of  12,000  tons  of  coal  arrived  at  Boston  on  3 
l);irges  and  2  schooners.  Each  schooner  held  3 J  times  as  much 
IS  each  barge.     Find  the  capacity  of  a  barge ;  of  a  schooner. 

4.  A  salmon  cannery  in  Alaska  paid  2^  for  each  red  fish 
tught  and  10^  for  each  king  salmon.     Two  men  brought  in 

.m;0  fish  one  day  and  received  $24.80  for  their  catch.     How 
many  fish  of  each  kind  did  they  catch  ? 

5.  The  powder  and  the  shell  uscil  in  a  twelve-inch  gun 
weigh  1265  pounds.  The  powder  weighs  15  pounds  more  than 
\  as  much  as  the  shell.     Find  the  weight  of  each. 

6.  A  merchant  bought  62  barrels  of  flour,  part  at  $  4J  per 
barrel,  the  rest  at  $5^  per  barrel.  If  he  paid  $320  for  the 
flour,  how  many  barrels  of  each  grade  did  he  buy  ? 

7.  During  a  year  of  365  days  one  locality  had  6  days  lesS 
nf  "clear"  weather  than  of  "cloudy"  weather,  and  4  days 
more  of  "  clear  "  than  of  "  partly  cloudy  "  weather.  Find  the 
number  of  days  of  each  kind  of  weather  during  the  year. 

8.  The  bark  from  a  cork  tree  lost  ^  of  its  weight  by  being 
hoiled.  The  boiled  cork  was  then  scraped,  its  weight  thus 
being  reduced  J.  How  much  did  the  cork  weigh  before  and 
after  these  two  operations,  if  the  entire  loss  was  16  pounds  ? 


164  SIMPLE   EQUATIONS 

9.  An  acre  of  wheat  yielded  2000  pounds  more  of  straw 
than  of  grain.  The  weight  of  the  grain  was  .3  of  the  total 
weight.  How  many  60-poLind  bushels  of  wheat  were  pro- 
duced ? 

10.  The  precious  stones  imported  into  this  country  one  year 
were  valued  at  $  40,000,000.  The  uncut  diamonds  were  worth 
f  as  much  as  the  cut  diamonds  but  twice  as  much  as  the  other 
precious  stones.     Find  the  value  of  each  kind  of  diamonds. 

11.  The  cost  per  mile  of  running  a  train  was  14^  less  with 
electrical  equipment  than  with  steam,  or  f  as  much.  What 
was  the  cost  per  mile  with  electricity  ? 

12.  A  Chicago  merchant  paid  $43.75  to  keep  7000  pounds 
of  butter  in  storage  for  4  months.  For  each  of  the  last  3 
months  he  was  charged  J  as  much  as  for  the  first  month. 
What  was  the  charge  per  pound  for  the  first  month,  and  for 
each  succeeding  month  ? 

13.  A  newspaper  reporter  saved  \  of  his  weekly  salary,  or 
$  1  more  than  was  saved  by  an  artist  on  the  same  paper,  whose 
salary  was  $5  greater  but  who  saved  only  ^  of  it.  How  much 
did  the  reporter  earn  per  week  ?  the  artist  ? 

14.  A  field  is  twice  as  long  as  it  is  wide.  By  increasing 
its  length  20  rods  and  its  width  30  rods,  the  area  will  be 
increased  2200  square  rods.     What  are  its  dimensions  ? 

15.  In  a  purse  containing  $1.45  there  are  J  as  many  quarters 
as  5-cent  pieces  and  |  as  many  dimes  as  5-cent  pieces.  How 
many  pieces  are  there  of  each  kind  ? 

16.  Find  a  fraction  whose  value  is  ^  and  whose  denomina- 
tor is  15  greater  than  its  numerator. 

17.  Find  a  fraction  whose  value  is  |  and  whose  numerator 
is  3  greater  than  half  of  its  denominator. 

18.  The  numerator  of  a  certain  fraction  is  8  less  than  the 
denominator.  If  each  term  of  the  fraction  is  decreased  by  5, 
the  resulting  fraction  equals  ^.     What  is  the  fraction  ? 


SIMPLE  EQUATIONS  165 

19.  An  experienced  woman  reeled  .12  Kg.  more  silk  per  day 
tioiii  ccxjoons  than  a  beginner.  During  one  week  (6  days)  she 
liad  J  of  a  day  lost  time,  yet  she  reeled  .36  Kg.  more  than  the 
beginner.     How  much  did  each  reel  per  day  ? 

20.  A  can  do  a  piece  of  work  in  8  days.  If  B  can  do  it  in 
10  days,  in  how  many  days  can  both  working  together  do  it? 

Solution 
Let  X  =  the  required  number  of  days. 

Then,  -  =  the  part  of  the  work  both  can  do  in  1  day, 

X 

\  —  the  part  of  the  work  A  can  do  in  1  day, 
^  =  the  part  of  the  work  B  can  do  in  1  day  ; 

a;     8      10'       40 

Solving,        x  =  Y»  or  4|,  the  required  number  of  days. 

21.  A  can  do  a  piece  of  work  in  10  days,  B  in  12  days,  and 
(J  in  8  days.     In  how  many  days  can  all  together  do  it? 

22.  A  can  pave  a  walk  in  6  days,  and  B  can  do  it  in  8  days. 
How  long  will  it  take  A  to  finish  the  job  after  both  have 
worked  3  days  ? 

23.  A  can  build  a  wall  in  15  days,  but  with  the  aid  of  B 
and  C  the  wall  can  be  built  in  6  days.  If  1^  does  as  much 
work  in  1  day  as  C  does  in  2  days,  in  how  many  days  can  B 
and  C  separately  build  the  wall  ? 

24.  A  and  B  can  dig  a  ditch  in  10  days,  B  and  C  can  dig  it 
in  ()  days,  and  A  and  C  in  7^  days.  In  what  time  can  each 
man  do  the  work  ? 

Suggestion.  — Since  A  and  B  can  dig  ^^  of  the  ditch  in  1  day,  B  and  C 
\  of  it  in  1  day,  and  A  and  C  f^  of  it  in  1  day,  tV  +  i  +  1^5  is  ^^i^e  the 
part  they  can  all  dig  in  1  day. 

25.  A  and  B  can  load  a  car  in  3  hours,  B  and  C  in  2\ 
hours,  and  A  and  C  in  2i  hours.  How  long  will  it  take  each 
alone  to  load  it  ? 


166  simplp:  equations 

26.  The  units'  digit  of  a  number  expressed  by  two  digits 
exceeds  the  tens'  digit  by  5.  If  the  number  increased  by  63 
is  divided  by  the  sum  of  its  digits,  the  quotient  is  10.  Find 
the  number. 

Solution 

Let  X  =  the  digit  in  tens'  place. 

Then,  x+  5  =  the  digit  in  units'  place, 

and  10  X  +  (x  +  5)  =  the  number ; 

...    10  a?  +  (a;  +  5)  -f  63  _  ^^  . 
2  X  +  5 
whence,  x  =  2, 

and  x  +  5  =  7. 

Therefore,  the  number  is  27. 

27.  The  tens'  digit  of  a  number  expressed  by  two  digits  is 
3  times  the  units'  digit.  If  the  number  diminished  by  33  is 
divided  by  the  difference  of  the  digits,  the  quotient  is  10. 
What  is  the  number  ? 

28.  The  tens'  digit  of  a  number  expressed  by  two  digits  is 
^  of  the  units'  digit.  If  the  number  increased  by  27  is  divided 
by  the  sum  of  its  digits,  the  quotient  is  6^.  What  is  the 
number  ? 

29.  A  girl  found  that  she  could  buy  18  apples  with  her 
money  and  have  5  cents  left,  or  12  oranges  and  have  11  cents 
left,  or  8  apples  and  6  oranges  and  have  10  cents  left.  How 
much  money  had  she  ? 

30.  An  officer,  attempting  to  arrange  his  men  in  a  solid 
square,  found  that  with  a  certain  number  of  men  on  a  side  he 
had  34  men  over,  but  with  one  man  more  on  a  side  he  needed 
35  men  to  complete  the  square.     How  many  men  had  he  ? 

Suggestion.  — With  x  men  on  a  side,  the  square  contained  x'^  men  ; 
with  a;  +  1  men  on  a  side,  there  were  places  for  (x  +  1)2  men. 

31.  A  regiment  drawn  up  in  the  form  of  a  solid  square  was 
reenforced  by  240  men.  When  the  regiment  was  formed  again 
in  a  solid  square,  there  were  4  more  men  on  a  side.  How 
many  men  were  there  in  the  regiment  at  first  ? 


SIMPLE   EQUATIONS  167 

32.  Mr.  Reynolds  invested  $800,  a  part  at  69^,  the  rest  at 
.")%.  The  total  annual  interest  was  $45.  Find  how  much 
money  he  invested  at  each  rate. 

Suggestion.  —  Let  x  =  the  number  of  dollars  invested  at  6%. 
Then,  800  —  x  =  the  number  of  dollars  invested  at  o  %  ; 

.-.   Tb«  +  T*ff(800-x)=45. 

33.  A  man  has  |  of  his  property  invested  at  4  %,  \  at  3  %, 
and  the  remainder  at  2%.  How  much  is  his  property  valued 
at,  if  his  annual  income  is  $  860  ? 

34.  A  man  put  out  $  4330  in  two  investments.  On  one  of 
them  he  gained  12  %,  and  on  the  other  he  lost  5  %.  If  his  net 
gain  was  $  251,  what  was  the  amount  of  each  investment  ? 

35.  Mr.  Bailey  loaned  some  money  at  4  %  interest,  but  re- 
ceived $  48  less  interest  on  it  annually  than  Mr.  Day,  who  had 
loaned  J  as  much  at  6%.     How  much  did  each  man  loan? 

36.  A  man  paid  $  80  for  insuring  two  houses  for  $  6000  and 
$  4000,  respectively.  The  rate  for  the  second  house  was  ^  % 
greater  than  that  for  the  first.    What  were  the  two  rates  ? 

37.  Mr.  Barnes  received  a  yearly  income  of  7%  from  an 
investment.  He  borrowed  twice  as  much  as  he  already  had 
invested,  paying  5  %  interest,  put  this  sum  with  his  original  in- 
vestment, and  then  received  a  net  income  of  $  385.  What  was 
the  sum  first  invested  and  the  sum  borrowed  ? 

38.  A  man  bought  some  50-dollar  shares  in  one  stock  com- 
pany and  I  as  many  100-dollar  shares  in  another.  At  the  end 
of  the  first  quarter,  dividends  of  2  %  and  of  1^  %,  respectively, 
were  declared  on  these  stocks,  and  the  man  received  $  120.  How 
much  money  did  he  invest  in  each  company  ? 

39.  My  deposit  in  a  savings  bank  that  pays  4%  interest 
is  6  times  as  great  as  my  deposit  with  a  trust  company.  On  the 
latter  I  receive  no  interest,  but  would  receive  3  %  interest  if  the 
deposit  were  $  300  or  more.  If  I  transfer  $  400  to  the  trust 
company,  my  interest  income  for  the  next  quarter  will  be  in- 
creased $  \.     Find  my  present  deposit  in  each  place. 


168  SIMPLE   EQUATIONS 

40.  At  what  time  between  5  and  6  o'clock  will  the  hands  of 
a  clock  be  together  ? 

Solution 

Starting  with  the  hands  in  the  position  shown, 
at  5  o'clock,  let  x  represent  the  number  of  minute 
spaces  passed  over  by  the  minute  hand  after  5 
o'clock  until  the  hands  come  together.  In  the 
same  time  the  hour  hand  will  pass  over  J^  of  x 
minute  spaces. 

Since  they  are  25  minute  spaces  apart  at  5  o'clock, 

X  _  A  ^  25  ; 
12 

.'.  X  =  27y\,  the  number  of  minutes  after  5  o'clock. 

41.  At  what  time  between  1  and  2  o'clock  will  the  hands  of 
a  clock  be  together  ? 

42.  At  what  time  between  6  and  7  o'clock  will  the  hands  of 
a  clock  be  together  ? 

43.  At  what  time  between  10  and  11  o'clock  will  the  hands 
of  a  clock  point  in  opposite  directions  ? 

44.  At  what  two  different  times  between  4  and  5  o'clock  will 
the  hands  of  a  clock  be  15  minute  spaces  apart  ? 

45.  A  man  rows  downstream  at  the  rate  of  6  miles  an  hour 
and  returns  at  the  rate  of  3  miles  an  hour.  How  far  down- 
stream can  he  go  and  return  within  9  hours  ? 

46.  A  motor  boat  went  up  the  river  and  back  in  2  hours  56 
minutes.  Its  rate  per  hour  was  17|-  miles  going  up  and  21 
miles  returning.     How  far  up  the  river  did  it  go  ? 

47.  An  express  train  whose  rate  is  40  miles  an  hour  starts  1 
hour  and  4  minutes  after  a  freight  train  and  overtakes  it  in 
1  hour  a,nd  36  minutes.  How  many  miles  does  the  freight 
train  run  per  hour? 

48.  A  yacht  goes  5  miles  downstream  in  the  same  time  that 
it  goes  3  miles  upstream ;  but  if  its  rate  each  way  is  dimin- 
ished 4  miles  an  hour,  its  rate  downstream  will  be  twice  its 
rate  upstream.     How  fast  does  it  go  in  each  direction  ? 


SIMPLE   EQUATIONS  169 

49.  The  distance  by  canal  from  Albany  to  Syracuse  is  166 
miles.  A  canal  boat  leaves  Albany  for  Syracuse,  moving  at 
the  rate  of  3  miles  in  2  hours ;  at  the  same  time  another  leaves 
Syracuse  for  Albany,  moving  at  the  rate  of  5  miles  in  4  hours. 
How  fur  from  Albany  do  they  meet  ? 

60.  The  distance  between  Southampton  and  New  York  is 
3046  nautical  miles,  or  knots.  A  vessel  left  Southampton 
for  New  York  and  sailed  at  the  rate  of  20  knots  an  hour.  15^ 
hours  later  another  vessel  started  from  New  York  and  sailed 
over  the  same  route  at  the  rate  of  18  knots  an  hour.  How  far 
from  Southampton  were  the  vessels  when  they  met  ? 

51.  At  3  P.M.  on  Monday  some  people  started  by  boat  from 
Toronto  for  Montreal,  where  they  remained  36  hours.  They 
returned  by  rail,  reaching  Toronto  at  4 :  30  p.m.,  Thursday. 
The  average  rate  was  13  miles  per  hour  by  boat  and  44  miles 
,)er  hour  by  rail.  The  distance  was  60  miles  greater  by  boat 
than  by  rail.     What  were  the  distance  and  the  time  each  way  ? 

52.  It  took  ^  passenger  train,  175  feet  long,  7^  seconds  to 
pass  completely  a  freight  train,  485  feet  long,  moving  in  the 
opposite  direction.  If  the  passenger  train  was  going  3  times  as 
fast  as  the  freight  train,  find  the  rate  of  each  per  hour. 

53.  In  making  5000  pounds  of  brass  there  were  used  8^ 
times  as 'much  copper  as  tin,  and  twice  as  much  tin  as  zinc. 
How  many  pounds  of  each  metal  were  used  ? 

54.  In  a  quantity  of  gunpowder  the  niter  composed  10 
pounds  more  than  J  of  the  weight,  the  sulphur  3  pounds  more 
than  ,'.y  of  it,  and  the  charcoal  3  pounds  less  than  ^  of  the 
weight  of  the  niter.     What  was  the  weight  of  the  gunpowder  ? 

55.  \jnited  States  silver  coins  are  ^\  pure  silver,  or  '  -j%  fine.' 
How  much  pure  silver  must  be  melted  with  250  ounces  of 
silver  J  fine  to  render  it  of  the  standard  fineness  for  coinage  ? 

56.  In  an  alloy  of  90  ounces  of  silver  and  copper  there  are 
()  ounces  of  silvc^r.  How  much  copper  must  be  added  that  10 
ounces  of  the  new  alloy  may  contain  J  of  an  ounce  of  silver  ? 


170  SIMPLE   EQUATIONS 

57.  If  80  pounds  of  sea  water  contain  4  pounds  of  salt,  how 
much  fresh  water  must  be  added  that  49  pounds  of  the  new 
solution  may  contain  1|  pounds  of  salt? 

58.  In  an  alloy  of  75  pounds  of  tin  and  copper  there  are  12 
pounds  of  tin.  How  much  copper  must  be  added  that  the  new 
alloy  may  be  12^  %  tin  ? 

59.  If  in  60  pounds  of  a  solution  of  salt  and  water  there  are 
3  pounds  of  salt,  how  much  fresh  water  must  be  evaporated 
from  the  solution  that  2o  pounds  of  the  new  solution  shall  con; 
tain  2|-  pounds  of  salt  ? 

60.  Of  24  pounds  of  salt  water,  12  %  is  salt.  In  order  to 
have  a  solution  that  shall  contain  4  %  salt,  how  many  pounds 
of  pure  water  should  be  added  ? 

61.  It  is  desired  to  add  sufficient  water  to  6  gallons  of 
alcohol  95%  x^ure  to  make  a  mixture  75  %  pure.  How  many 
gallons  of  water  are  required  ? 

62.  Four  gallons  of  alcohol  90  %  pure  is  to  be  made  50  % 
pure.     What  quantity  of  water  must  be  added  ? 

63.  A  body  placed  in  a  liquid  loses  as  much  weight  as  the 
weight  of  the  liquid  displaced.  A  piece  of  glass  having  a 
volume  of  50  cubic  centimeters  weighed  94  grams  in  air  and 
51.6  grams  in  alcohol.  How  many  grams  did  the  alcohol 
weigh  per  cubic  centimeter? 

64.  Brass  is  8f  times  as  heavy  as  water,  and  iron  7^  times 
as  heavy  as  water.  A  mixed  mass  weighs  57  pounds,  and 
when  immersed  displaces  *7  pounds  of  water.  How  many 
pounds  of  each  metal  does  the  mass  contain? 

Suggestion.  —  Let  there  be  x  pounds  of  brass  and  (57  —  x)  pounds 
of  iron.     Then,  x  pounds  of  brass  will  displace  (x  -^  8f )  pounds  of  water. 

65.  If  1  pound  of  lead  loses  2%  of  a  ^Dound,  and  1  pound  of 
iron  loses  ^V  ^^  ^  pound  when  weighed  in  water,  how  many 
pounds  of  lead  and  of  iron  are  there  in  a  mass  of  lead  and  iron 
that  weighs  159  pounds  in  air  and  143  pounds  in  water  ? 


SIMPLE  EQUATIONS  171 

66.  If  tin  and  lead  lose,  respectively,  ^  and  -^  of  their 
weights  when  weighed  in  water,  and  a  60-pound  mass  of  lead 
and  tin  loses  7  pounds  when  weighed  in  water,  find  the  weight 
of  the  tin  in  this  mass. 

67.  If  97  ounces  of  gold  weighs  92  ounces  when  it  is  weighed* 
in  water,  and  21  ounces  of  silver  weighs  19  ounces  when  it  is 
weighed  in  water,  how  many  ounces  of  gold  and  of  silver  are 
there  in  a  mass  of  gold  and  silver  that  weighs  320  ounces  in 
air  and  298  ounces  in  water  ? 

68.  If  zinc  weighs  437.5  pounds  per  cubic  foot  and  copper 
r)50  pounds,  what  per  cent  by  volume  is  each  of  these  metals 
in  an  alloy  of  them,  1  cubic  foot  of  which  weighs  532  pounds  ? 

Solution  of  Formulae 

236.  A  formula  expresses  a  principle  or  a  rule  in  symbols. 
The  solution  of  problems  in  commercial  life,  and  in  mensura- 
tion, mechanics,  heat,  light,  sound,  electricity,  etc.,  often  de- 
pends upon  the  ability  to  solve  formulae. 

EXERCISES 

237.  1.  The  circumference  of  a  circle  is  equal  to  tr  (=3.1416) 
times  the  diameter,  or  C  =  7rD 

Solve  the  formula  for  D  and  find,  to  the  nearest  inch,  tlie 

(lianieter  of  a  locomotive  wheel  whose  circumference  is  194.78 

inches. 

Solution 

From  C  -  nD,  wD  =  C. 

.'.  D  =  ^=^^i^  =  62.0+. 
T      3.1416 

Hence,  to  the  nearest  inch,  the  diameter  is  62  inches. 
2.   Area  of  a  triangle  =  ^  (base  x  altitude),  or 

Solve  fur  h,  ilieii  tiiid  the  base  of  a  triangle  whose  area  is  600 
sfjuare  feet  and  altitude  4U  feet. 


172  SIMPLE   EQUATIONS 

3.  The  area  of  a  trapezoid  is  equal  to  the  product  of  half 
the  sum  of  the  bases  and  the  altitude ;  that  is, 

The  bases  are  h  and  b' ;  b'  is  read  '  6-pnme.' 

Solve  for  h,  then  find  the  altitude  of  a  trapezoid  whose  area  is 
96  square  inches  and  whose  bases  are  14  inches  and  10  inches, 
respectively. 

4.  The  volume  of  a  pyramid  =  i  (base  x  altitude),  or 

Solve  for  B,  then  find  the  area  of  the  base  of  a  pyramid  whose 
volume  is  252  cubic  feet  and  altitude  9  feet. 

5.  The  charge  (c)  for  a  telegram  from  ISTew  York  to  Chicago, 
40^  for  10  words  and  3^  for  each  additional  word,  may  be 
found  by  the  formula, 

c  =  40  +  3  (/J -10), 
in  which  n  stands  for  the  number  of  words. 
Find  the  cost  of  a  16-word  message. 
Solve  for  n,  then  find  how  many  words  can  be  sent  for  $  1. 

6.  In  the  formula         '  =  P  '  r^  •  '> 

i  denotes  the  interest  on  a  principal  of  p  dollars  at  simple 
interest  at  r  %  for  t  years. 

Solve  for  t,  then  find  the  time  ^  300  must  be  on  interest  at 
5%  to  yield  $60  interest. 

Solve  for  r.  At  what  rate  of  interest  will  $4500  yield  $  900 
interest  in  5  years  ? 

Solve  for  p.  AVhat  principal  at  3^%  will  yield  $210 
annually  ? 

7.  In  §  42  was  given  the  formula  for  the  space  (s)  passed 
over  by  anything  that  moves  with  uniform  velocity  (y)  during 
a  given  time  {t).     It  is 

5  =  vf. 
Solve  for  -y,  then  find  the  velocity  of  sound  when  the  condi- 
tions are  such  that  it  travels  8640  feet  in  8  seconds. 


SIMPLE   EQUATIONS  178 

8.  The  formula  for  converting  a  temperature  of  F  degrees 
Falirenheit  into  its  equivalent  temperature  of  C  degrees  Centi- 
grade is 

C  =  \{F-Z2). 

Solve  for  F  and  express  25°  Centigrade  (the  mean  annual 
temperature  in  Havana)  in  degrees  Fahrenheit. 

9.  If  a  steel  rail  at  0°  C.  is  heated,  for  every  degree  it  is 
heated  it  will  expand  a  certain  part  of  its  original  length.  Let 
Zo  (read,  *  L  sul>zero ')  denote  its  original  length  at  0°  C,  L  its 
length  at  t  degrees  C,  and  a  the  certain  fractional  multiplier, 
or  'coefficient  of  expansion.' 

Then,  L^L^+U-at. 

Solve  for  a.  A  steel  rail  30  feet  long  at  0°  C.  expanded  to 
a  length  of  30.001632  feet  at  50°  C.     Find  the  value  of  a. 

Solve : 

10.  s  =  )j  at\  for  a.  15.    Mvi  =  mv^,  for  m. 

11.  F=3/a,  fora.  16.  E  ==  ^  Mn',  fov  M. 

12.  F  =fjLW,  for  fx,  17.  PoVo  =  rV,fovP. 

13.  ir=  Fs,  for  8,  18.  s  =  V(/  -f  ^  at'^,  for  a. 

14.  P=PR,iov  R.  19.  8  =  \a{2t-\\iovt. 

20.  Any  sort  of  a  bar  resting  on  a  fixed  point  or  edge  is 
called  a  lever ;  the  point  or  edge  is  called  the  fulcrum. 

A  lever  will  just  balance   when 

the  numerical  product  of  the  power      (p\ p        ^ 

(p)  and  its  distance  (d)  from  the      T  d  ▲  ^     T 

fulcrum  (F)  is  equal  to  the  numeri- 
cal product  of  the  weight  (  W)  and  its  distance  (D)  from  the 
fulcrum ;  that  is,  when 

pd=WD. 

Solve  for  TTand  find  what  weight  a  power  of  150  (pounds) 
will  support  by  means  of  the  lever  shown,  if  d  =  7  (feet)  and 
Z)  =  3  (feet). 


174  SIMPLE   EQUATION'S 

Find  for  what  values  of  p,  d,  W,  or  D  the  following  levers 
will  balance,  each  lever  being  8  feet  long : 

21.    ? , ?-^44  23.  6QQ  3    ^ ^ 


p 

i^       144 

300 

G 

100 

oo       300       i'^ 100  ^A      ^t  F 

700 

25.  Philip,  who  weighs  114  pounds,  and  William,  who 
weighs  102  pounds,  are  balanced  on  the  ends  of  a  9-foot  plank. 
]N"eglecting  the  weight  of  the  plank,  how  far  is  Philip  from  the 
fulcrum  ? 

26.  The  figure  illustrates  the  lever  of  a  safety  valve,  the 

power  being  the  steam  pressure 
^  z>  =  50  —>  (»)  acting  on  the  end  of  the  pis- 

ton  above.  The  area  of  the 
end  of  the  piston  is  16  square 
inches.  What  weight  (1^  must 
be  hung  on  the  end  of  the  lever 
'steamVe'isiire =p i;  ^"-^' i'-^.''^\  SO  that  whcu  the  stcam  pres- 
sure rises  to  100  pounds  per 
square  inch  the  piston  will  rise  and  allow  steam  to  escape? 

Solve: 

27.  —  =  ^,  for  R.  29.    a  =  ^^^Z^,  for  v^. 


w 


'^S'j.ssyt^sssssvsx^^sysyssssssssssx's'^ 


E'     H' 
E 


28.    1=  -^—,  for  r.  30.    F=  Fo(^l  +^\  for  t. 


31. 

Solve  ^='^  +  '*,  for^;  f or  r. 
e           r 

32. 

Solve  i.  =  14-1,  for/.;  for/,. 

33. 

Solve  \  Wl  =~~,  for  W;  for  ^ 

3T  —  . 
C  C 

34.    Solve  C=  -^  +  i,  for  C\',  for  C^. 
Ci      62 


SIMPLE   EQUATIONS  175 

35.  The  number  of  pounds  pressure  (P)  on  A  square  feet 
of  surface  of  any  body  submerged  to  a  depth  of  h  feet  in  a 
liquid  that  weighs  w  pounds  jx  r  (  iibic  foot  is  given  by  the 
formula  /» =  *^A. 

Fresh  water  weighs  about  62^  pounds  per  cubic  foot,  and  ordinary  sea 
water  about  04  pounds  per  cubic  foot. 

Find  the  pressure  on  1  square  foot  of  surface  at  the  bottom 
of  a  standpipe  in  which  the  water  is  30  feet  high;  at  the 
bottom  of  the  ocean  at  a  depth  of  3000  feet. 

36.  Solve  P=wAh  for  h  and  find  the  value  of  h  when 
P  =  5000,  w  =  62^,  and  ^  =  8. 

37.  At  what  depth  in  fresh  water  will  the  pressure  on  an 
object  having  a  total  area  of  4  square  feet  be  2000  pounds  ? 

38.  The  bottom  of  a  rectangular  cistern  is  0  feet  square. 
For  what  depth  of  water  will  the  pressure  on  the  bottom 
l)e  36,000  pounds  ? 

39.  How  deep  in  the  ocean  can  a  diver  go,  without  danger,  in 
a  suit  of  armor  that  can  sustain  safely  a  pressure  of  140 
pounds  per  square  inch  (20,160  pounds  per  square  foot)  ? 

40.  If  the  pressure  per  square  foot  on  the  bottom  of  a  tank 
holding  18  feet  of  petroleum  is  990  pounds,  what  is  the  weight 
of  the  petroleum  per  cubic  foot  ? 

41.  The  side  of  a  chest  lying  in  25  feet  of  water  was  5 
square  feet  in  area  and  sustained  a  pressure  of  8000  pounds. 
Was  the  chest  submerged  in  fresh  or  in  salt  water  ? 

42.  riif  pressure  on  the  inner  surface  of  a  water  pipe  is  60 
pounds  per  square  inch  at  the  faucet  in  the  basement  of  a 
house  and  40  pounds  per  square  inch  at  the  faucet  in  the 
top  story.  How  much  higher  is  the  faucet  in  the  top  story 
than  the  one  in  the  basement  ? 

Sdggestion.  —  The  pressure  due  alone  to  the  height  of  the  upper  faucet 
above  the  lower  one  is  60  pound.**  less  40  pounds,  or  20  pounds,  per  square 
inch,  or  2880  pounds  per  square  foot. 


SIMULTANEOUS   SIMPLE   EQUATIONS 


TWO    UNKNOWN   NUMBERS 

238.  In  the  equation  x  -\-  y  =  12, 

X  and  y  may  have  an  unlimited  number  of  pairs  of  values, 

as  X  =  1  and  y  =  ll; 

or  X  =  2  and  2/  =  10 ;  etc. 

For  since  y  =  12  —  x, 

if  any  value  is  assigned  to  x,  a  corresponding  value  of  y  may  be 
obtained. 

An  equation  that  is  satisfied  by  an  unlimited  number  of  sets 
of  values  of  its  unknown  numbers  is  called  an  indeterminate 
equation. 

239.  Principle.  —  Any  sirigle  equation  involving  two  or  more 
unknoicn  numbers  is  indeterminate. 

240.  The  equations     2x-\-2y  =  10 
and  3x  -\-  3y  =  15 
express  but  one  relation  between  x  and  ?/;    namely,  that  their 
sum  is  5.     In  fact,  the  equations  are  equivalent  to 

x  +  y  =  5 
and  to  each  other.     Such  equations  are  often  called  dependent 
equations,  for  either  may  be  derived  from  the  other. 

241.  The  equations       x  -{-  y  =  5) 

X  —  y  =  1 } 
express  two  distinct  relations  between  x  and  y,   namely,  that 


176 


SIMULTANEOUS  SIMPLE   EQUATIONS  177 

t  heir  sum  is  5  and  their  difference  is  1.    The  equations  cannot  be 
reduced  to  the  same  equation ;  that  is,  they  are  not  equivalent. 
Equations  that  express  different  relations  between  the  un- 
known numbers  involved,  and   so  riuuiot  be  reduced  to  the 
iiiit'  (Miiiat  inii,  are  called  independent  equations. 

242.    Each  of  the  equations 


x-\-y  =  5] 


s  satisfied  separately  by  an  unlimited  number  of  sets  of  values 

t  X  and  2/,  but  they  have  only  one  set  of  values  in  common, 

namely, 

X  =  S  and  t/  =  2. 

Two  or  more  equations  that  are  satisfied  by  the  same  set  or 
st'ts  of  values  of  the  unknown  numbers  form  a  system  of 
simultaneous,  or  consistent,  equations. 

243.    The  equations 


x-\-y  =  5) 
x  +  y  =  7} 


have  no  set  of  values  of  x  and  y  in  common. 
Such  equations  are  called  inconsistent  equations. 

244.  The  student  is  familiar  with  the  methods  of  solving 
simple  equations  involving  one  unknown  number.  The  gen- 
eral method  of  solving  a  system  of  two  independent  simul- 
taneous simple  equations  in  tivo  nuloioun  nmnlxns, 

as  x-^y  =  d\ 

x-^y  =  S) 

>  to  combine  the  equations,  using  axioms  1-4  in  such  a  way 
as  to  obtain  an  equation  involving  x  alone,  and  another  in- 
volving y  alone,  which  may  be  solved  separately  by  previous 
methods. 

The  process  of  deriving  from  a  system  of  simultaneous  equa- 
tions  another  system  iiivnlvinij  fewer  unknown  n\iiiibers  is 
called  elimination. 

MIL.NE^B    8TAN1>.    ALO. 12 


178  SIMULTANEOUS   SIMPLE    EQUATIONS 

Elimination  by  Addition  or  Subtraction 

245.  In  solving  simultaneous  equations  we  may  apply  axi- 
oms 1-4,  subject  to  the  restrictions  mentioned  in  §  230  in 
regard  to  the  introduction  or  removal  of  roots  in  multiplying 
or  dividing  both  members  by  expressions  involving  unknown 
numbers. 

5x-{-2y  =  24.  5x  +  2y  =  2i 

5x-2y  =  16  5x-2y  =  16 

Adding,  10  cc         =40        Subtracting,  4?/=  8 

In  the  two  given  equations  the  coefficients  of  y  are  numeric- 
ally equal  and  opposite  in  sign.  Therefore,  if  the  equations 
are  added  (Ax.  1),  the  resulting  equation  will  not  involve  y. 
This  method  of  eliminating  y  illustrates  elimination  by  addition. 

If  one  equation  is  subtracted  from  the  other  (Ax.  2),  the 
resulting  equation  will  not  involve  x.  The  second  process 
illustrates  elimination  by  subtraction. 

EXERCISES 

246.  1.    Solve  the  equations  2  ic -h  3  2/ =  7  and  3  a;  H- 4  2/ =  10. 

Solution 

2x+    31/ =  7,  (1) 

3x+    ^y  =  10.  (2) 

(1)X4,  Sx  +  V2y=2S.  (3) 

(2)  X  3,  9x-\-  12?/ =  30.  (4) 

(4) -(3),  x=2.  (5) 

Substituting  (5)  in  (1),  4  +    3  ?/  =  7. 

.-.  y  =  l. 
To  verify,  substitute  2  for  x  and  1  for  y  in  the  given  equations. 

Rule.  —  If  necessary,  multiply  or  divide  the  equations  hy  such 
numbers  as  will  7nake  the  coefficients  of  the  quantity  to  be  elimi- 
nated numerically  equal. 

Eliminate  by  addition  if  the  resulting  coefficieyits  have  milike 
sig7is,  or  by  subtraction  if  they  have  like  signs. 


SIMULTANEOUS  SIMPLE   EQUATIONS 

Solve  by  addition  or  subtraction,  and  verify  results : 

l2a;4-5y  =  47. 


179 


3. 


4. 


7. 


8. 


3x-iy  =  7, 
[x  +  10y  =  25. 

f2x-10.y  =  15, 
2x-4y  =  lS. 

3w-v  =  4, 
.M+3v=-2. 

x  +  Sy  =  21. 

2a-f.'U  =  17, 
[3a  +  26  =  18. 


(  7  s  —  9  V  —  6, 
1  8  -h  2  y  =  14. 


11. 


12. 


13. 


13^-M  =  20, 
.4«4-2m  =  20. 

3dH-4y  =  25, 
.4d4-3y  =  31. 

r5/)  +  6^  =  32, 
17^-39  =  22. 


(  3 rt  +  6z=: 39, 
I9a-42  =  51. 


^g     f8x-3y  =  44, 
\7x-5y  =  29. 


16. 


6a;-5y  =  33, 
4  a;  4-  4  y  =  44. 


j^      f3m  +  ll7i==67 
I  5  m  —  3  n  =  5. 


Elimination  by  Comparison 
247.    If  x  =  S-y,  (1) 

and  also  a;  =  2  -|-  ?/,  (2) 

by  axiom  5,  the  two  expressions  for  x  must  be  equal. 
.•.8-i/  =  2  +  y. 

By  comjmring  the  vahies  of  x  in  the  given  equations,  (1)  and 
(2),  we  have  eliminated  x  and  obtained  an  equation  involving 
//  alone. 

This  method  is  called  elimination  by  comparison. 


180 


SIMULTANEOUS   SIMPLE   EQUATIONS 


EXERCISES 

248.    1.    Solve  the  equations  2x  —  3y  =  10  and  5x-\-2y  =  (J. 
Solution 


{2x-Sy  =  lO, 


5x  +  2//  =  (i 


From  (1), 
From  (2), 


io  +  ''^y 


'^y. 


(1) 

(2) 
(4) 


Comparing  the  values  of  x  in  (3)  and  (4), 

10  +  o  ■?/  ^  t>  -  2  y  _ 

2  5*  • 

Solving,  y  =  —  2. 

Substituting  —  2  for  ?/  in  either  (3)  or  (4), 

x  =  2. 
To  vetify,  substitute  2  for  x  and  -  2  for  ?/  in  the  given  equations. 

Rule.  —  Find  an  expression  for  the  value  of  the  same  vnknown 
nuinher  in.  each  equation,  equate  the  two  expressions,  and  solve  the 
equation  thus  formed. 

Solve  by  comparison,  and  verify  results: 


2.     \ 


{?,x-2y  =  10, 


3. 


6.     ^ 


[x-\-y  =  l(). 

5x  +  y  =  22, 
x-\-5y  =  14:. 

2x-{-3y  =  2^, 
5x-3y  =  lS. 

3  ic  +  5  ?/  =  14, 
2x-3y  =  3. 

(3v  +  2y  =  36, 


^^    (2s  +  7t  =  S, 
[32t--2v  =  l. 


9.    i 


r4'y  +  3w  =  34, 


10.    \ 


{llv-\-5iv  =  S7. 
ix-13y  =  5, 


[5v-9y  =  23, 


[3x-\-lly  =  -17. 

^^     |18a^-3^=.4^, 
11-4  a; +  3?/  =  27. 


SIMULTANEOUS  SIMPLE   EQUATIONS  181 

Elimination  by  Substitution 

249.  Given  :ix-\-'2  f/ =  27,  (1) 
and                                          X-    .y  =  4.  (2) 

On  solving  (2)  for  x,  its  vain.'  is  touiid  to  l.c  .c  =  4-|-i/. 

If  4  -h  >/  is  substituted  for  x  in  (1 ),  o.r  will  hecoine  3(4  4-y), 
and  the  regulting  equation 

3(4+y)  +  2;y  =  27  (3) 

will  involve  y  only,  x  having  been  eliminated. 

Solving  (3),  y  =  3. 

Substituting  3  for  y  in  (2),  x  =  7. 

This  method  is  called  elimination  by  substitution.      ^ 

Rule.  —  Find  an  expression  for  tjir  ndue  of  either  of  the  wn- 
known  numbers  in  (me  of  the  ef/'i<ifi"ii.s. 

Substitute  this  value  for  that  unknoum  number  in  the  other 
equation,  and  solve  the  resulting  etjuatioyi. 

EXERCISES 

250.  Solve  by  substitution,  and  verify  results : 
x  —  y=4,  fi7  =  Sx  +  Zj 

D. 


i;: 


[4y-x=U.  [7=Sz-2x. 

|ar  +  ?/  =  10,  ^      ay  =  10-x, 

[6x  —  7y  =  M.  \y  —  x  =  5. 


U-8y  =  22.  1: 


'3ic-4.v  =  26,  .      r72-3a;  =  18, 

22-5a;  =  l. 


f  6 1/- 10  a;  =14,  (3-15y=-x, 

y-x  =  3.  '  l34-15y  =  4x. 

^  +  l=3x,  ^^  rl-a;  =  3.y, 

5a;  +  9  =  3//.  '  [3(1  -  x)  =  iO  -  y. 


182 


SIMULTANEOUS  SlMl»LE  EQUATIONS 


251.  Three  standard  methods  of  elimination  have  been 
given.  Though  each  is  applicable  under  all  circumstances,  in 
special  cases  each  has  its  peculiar  advantages.  The  student 
should  endeavor  to  select  the  method  best  adapted  or  to  invent 
a  method  of  his  own. 

EXERCISES 

252.  Solve  by  any  method,  verifying  all  results : 


1. 


\x  —  z  =  5. 


5. 


x-h3  =  y-3, 
2(x  +  3)=6-y. 


3x-{-y  =  10, 
x-^3y  =  6. 


(5x-y  =  12, 
[x-\-3y  =  12. 


4:x  +  5y=  -2, 
5x-^4:y  =  2. 

(5x-y  =  2S, 
[3x  +  5y  =  2S. 


^      (H2-x)=3y, 

{2(2-x)  =  2(y-2). 

g     |(a.  +  l)  +  (2/_2)  =  7, 
l(a;  +  l)-(2/-2)  =  5. 


Eliminate  before  or  after  clearing  of  fractions,  as  may  be 
more  advantageous : 


^  +  ?:=n, 


5  +  32/  =  21. 


12. 


■y 


=  -2, 


oa-"  ,  .»/_io 


10. 


4        o 
3^5 


13. 


x-i-y      x-y 
3     "^     4 


11. 


11. 


3  2 


X      2y 
3       7 


8. 


14. 


^-^-1=0, 
2     3 

2a;-l      3v- 


SIMULTANEOUS  SIMPLK   EQUATIONS 


188 


15. 


16. 


17. 


18. 


3      3 


4        3 

^  +  ^  =  17. 
2      4 


x-\ 
4 

x-1 


4-y  =  3, 
+  42/  =  9. 


+  42,  =  15, 


6       3 


19. 


-A_  +  3  =  0. 

[35  —  2^ 

a; 


=0, 


20. 


-12 


//4-32 


21. 


22. 


2/     3x-2j^25. 
8^        5 

5  4 

5y-7     4x-3^^g_5^ 
2  6 

'  .2t/+.5  ^  .49  x-.l 

1.5  4.2      ' 

.bx-2  ^  41  _  1.5y-ll 

1.6  16  8 


23. 


24. 


25. 


26. 


a?  +  y  .  x-y_2x'-y     4y-x     ^ 

a;  =  2y. 

a;4.j(3a;-»/-l)  =  i+f(2/-l), 
K4a:-f3?/)=Ti,(7y  +  24). 

6 a; 4-9  ,  3x-{-oy      „,   ,  3a;  +  4 

8y-t-7     6a;-3y^,      4y-9 
10         2(y-4)  5 

3a;-5y     2g-8.v-9^31 
3  12  12' 

l(M  +  H)-(4x-|-25)  =  |. 


184 

27. 


SIMULTANEOUS   SIMPLE   EQUATIONS 

28.     P 

29. 


Kii-l)-ir  =  4i 


2.4  fi-. 04?^  =  .62, 
7  w  +  .15  d  =  1.795. 


30. 


)(3v  +  ,5)=6v(u 
2y  —  x      2  a;  —  59 


\(2u-t 


(2  u  +  .1)(3  v  +  .5)=6  v(u  +  .3) 
ic-20- 
2/ 


31. 


a;-18 
a?      16  — a; 


23 -a^ 

30  = 


3-?/       QA_3y-73 


2 

3" 

3Q     5.v  +  2a^ 
40 -a? 


4(^-6)     83-8^^^^  _ 
y  +  S    ^8  ^' 


Equations  of  the  form  -4--=c,  though  not  simple  equa- 

X      y 

tions,  may  be  solved  as  simple  equations  for  some  of  their 

roots  by  regarding  -  and  -  as  the  unknown  numbers. 
X  y 


32.    Solve  the  equations 


Solution.     (2)  — (1), 


Substituting  (3)  in  (1), 


From  (4)  and  (3), 


4     3      14 

X      y       5' 

4      10      50 
X      y       3* 

13      208 
y       15 

1_16 

"  y    15* 

4      48  _  H^ 
X      15       5* 

.     1_3 

X  =  -  and  y 

h 

_15 

16 

(1) 

(2) 
(3) 
(4) 


SIMULTANEOUS  SIMPLE   EQUATIONS 


185 


Solve,  and  verify  results: 

53^  _2 

X     y 


33. 


?5  +  l  =  6. 
X      y 


34.    I 


36.    { 


36. 


2 

3_ 

-  5. 

X 

y 

5 

_2_ 

=  7. 

a; 

y 

4 

,  3 

9 

+  -  = 

-  rt? 

X 

2/ 

8 

3 

,  4 

11 

+  -  = 

X 

y 

12 

'7 

4-?  = 

:30, 

a; 

;/ 

7 

^«= 

:30. 

y 

a; 

37. 


38. 


39. 


40.    { 


X 

6_ 
3/ 

64, 

X 

5  _ 

y~ 

73^. 

3 

2a; 

_i_ 

2/ 

=  -3, 

5 
[2  a: 

2^ 

=  23. 

10_^5_ 
a;       ?/ 

20, 

X 

10_ 

y 

57^. 

'7 
Sx 

2 
33/ 

=  10, 

5 
6a; 

2 
11 

-  =  17 
V 

Solve  the  following  as  if 


1 


1 


a;-l'2/+l 
numbers,  and  then  find  the  values  of  x  and  y : 


,  etc.,  were  the  unknown 


41. 


42. 


-?-  -f  -^  =  12. 
la;  — 1      y  +  1 

_5 3_ 

x-1     y-\ 

_2 1_ 

x—1    y—1 


=  14, 

=  6. 


43.    ^ 


44. 


y      2  — a; 

r>^_6_ 

?/      2-a; 


1 


H-9. 


.r      ?/  4-  3 

7  ^  _.'3_ 
a;     y  +  3 


-10. 


186 


SIMULTANEOUS   SIMPLE   EQUATIONS 


Literal  Simultaneous  Equations 

ax-\-by  =  m, 
GX  -\-dy=n. 


253.    1.  Solve  the  equations 


Solution 

ax-\-by  =  m 

ex  -\-  dy  =  n 

(1)  X  d, 

adx  +  hdy  —  dm 

(2)  X  6, 

hex  +  hdy  =  hn 

(3) -(4), 

{ad  —  hc)x  =  dm  - 

-  bn 

ad- 

-hn 
-he 

(1)  X  c, 

acx  +  hey  =  em 

(2)  X  a, 

acx  -\-  ady  =  an 

(7) -(6), 

(ad  —  he)y  —  an  - 

-  cm 

.•.^/  =  «^- 

em 

ad  —  be 


(1) 
(2) 
(3) 
(4) 


(5) 

(6) 
(7) 


(8) 


In  solving  literal  simultaneous  equations,  elimination  is  usually  per- 
formed by  addition  or  subtraction. 

Solve  for  x  and  y,  and  test  results  by  assigning  suitable 
values  to  the  other  letters : 


2. 


3. 


4. 


5. 


ax-\-by 
bx  —  ay 


m, 


c. 


ax—by  =  m, 
cx  —  dy  =  r. 

ax  =  by, 
X  -{-y  =  ab. 

mix  -{-y)  =  a, 
n(x  —  y)  =2  a. 


6. 


7. 


\a(x-y)  =  5, 
bx  —  cy  =  n. 

a{a  —  x)  =  b(y  —  b), 
ax  =  by. 

x-{-y  =  b-a, 

bx  —  ay  4-  2  ab  =  0. 

(x-y  =  a-b, 
[ax->rby  =  a^  —  b^. 


SIMULTANEOUS  SIMPLE   EQUATIONS 


187 


10. 


11. 


12. 


13. 


14. 


0 


x-fl      a-fft  +  l 


0. 


(I 

bx  —  ay 

X  y  a 
11^1 
X     y     b 


X     y 


=  -1. 


i«    y 


?•+  y 


2ab, 


y 


\  +  ^^a  +  b. 
{ ab     afj 

a     b 

6     a     2* 


15. 


16. 


17. 


18. 


19. 


20.    Given 


r  F=  Ma, 


y  +  i 

a  — 

b  +  1 

x  —  y  = 

2  b. 

r   1 

1 

x  —  a 

a- 

y 

x  +  y_ 
x-y 

■a. 

?J.2=, 


b     c 


X     y 
X      y 


ax     by 


[bx 


l-l  =  cZ. 


ay 


Find  the  values  of  F  and  a  when  3/  =  15,  s  =  72,  and  ^  =  6. 

l  =  a-\-{n  —  l)d, 


21.    Given 


«  =  |(«  +  0- 


Find  the  values  of  a  and  I  when  n  =  50,  d  =  2,  and  s  =  2500 ; 
the  values  of  d  and  a  when  Z  =  50,  n  =  25,  and  s  =  650. 


(I  =  a7^- 


22.    Given 


8  = 


rl  —  a 
r-1 


Find  the  values  of  a  and  Z  when  r  =  2,  n  =  11,  and  s  =  2047. 


188  SIMULTANEOUS   SIMPLE    EQUATIONS 

Problems 
254.    Find  two  numbers  related  to  each  other  as  follows : 

1.  Sum  =  14 ;  difference  =  8. 

2.  Sum  of  2  times  the  first  and  3  times  the  second  =  34  5 
sum  of  2  times  the  first  and  5  times  the  second  =  50. 

3.  Sum  =  18 ;  sum  of  the  first  and  2  times  the  second  =  20. 

4.  A  grocer  sold  2  boxes  of  raspberries  and  3  of  cherries  to 
one  customer  for  54^,  and  3  boxes  of  raspberries  and  2  of 
cherries  to  another  for  56 f^.     Find  the  price  of  each  per  box. 

5.  A  druggist  wishes  to  put  500  grains  of  quinine  into  3- 
grain  and  2-grain  capsules.  He  fills  220  capsules.  How  many 
capsules  of  each  size  does  he  fill  ? 

6.  On  the  Fourth  of  July,  850  glasses  of  soda  water  were 
sold  at  a  fountain,  some  at  5^  each,  the  others  at  10^  each. 
The  receipts  were  $55.     How  many  were  sold  at  each  price  ? 

7.  A  fruit  dealer  bought  36  pineapples  for  $2.50.  He 
sold  some  at  12^  each  and  the  rest  at  10^  each,  thereby  gain- 
ing $1.50.     How  many  did  he  sell  at  each  price  ? 

8.  The  receipts  from  300  tickets  for  a  musical  recital  were 
$125.  Adults  were  charged  50^  each  and  children  25^  each. 
How  many  tickets  of  each  kind  w^ere  sold  ? 

9.  An  errand  boy  went  to  the  bank  to  deposit  some  bills 
for  his  employer.  Some  of  them  were  1-dollar  bills,  and  the 
rest  2-dollar  bills.  The  number  of  bills  was  38  and  their  value 
was  $50.     Find  the  number  of  each. 

10.  If  2  full-grown  rubber  trees  in  Brazil  yield  4  pounds 
more  of  rubber  in  a  year  than  8  trees  in  Ceylon,  and  3  Brazil 
trees  yield  10  pounds  more  than  10  Ceylon  trees,  what  is  the 
average  yield  per  tree  in  each  country  ? 

11.  A  man  noticed  that  a  15-word  message  by  telegraph  cost 
him  40^  and  a  22-word  message  54/,  between  the  same  two 
cities.  Find  the  charge  for  the  first  ten  words  and  the  charge 
for  each  additional  word. 


SIMULTANEOUS  SIMPLE   EQUATIONS  189 

12.  At  a  factory  where  1000  men  and  women  were  employed, 
the  average  daily  wage  was  $2.50  for  a  man  and  $1.50  for  a 
woman.  If  labor  cost  $2340  per  day,  how  many  men  were 
•  inployed?  how  many  women? 

13.  It  reciuired  60  inches  of  tape  to  bind  the  four  edges  of 
:i  card  on  which  a  photograph  was  mounted.  The  length  of 
the  card  was  6  inches  greater  than  the  width.  How  many 
inches  long  was  the  card  ?  how  many  inches  wide  ? 

14.  A  lieutenant  of  the  U.  S.  navy  received  $150  per  month 
while  on  sea  duty  and  $127.50  per  month  while  on  shore  duty. 
I  lis  salary  for  a  year  amounted  to  $1620.  How  many  months 
was  he  on  sea  duty  ?  on  shore  duty  ? 

15.  The  great  columns  of  Bedford  stone  in  the  Indianapolis 
])ost  office  building  weigh  94  tons  each,  including  the  shafts 
and  the  capitals  resting  on  them.  Each  shaft  weighs  74  tons 
more  than  its  capital.     Find  the  weight  of  a  shaft ;  of  a  capital. 

16.  The  receipts  from  a  football  game  were  $  700.  Admis- 
sion tickets  to  the  grounds  were  sold  for  50  ^,  and  to  the  grand 

land,  for  25^  in  addition.  If  twice  as  many  persons  had 
purchased  tickets  for  the  grand  stand,  the  receipts  would  have 
been  $800.     How  many  tickets  of  each  kind  were  sold? 

17.  The  duty  paid  on  an  importation  of  40,000  shingles 
and  160,000  laths  was  $52,  and  that  on  80,000  shingles  and 
70,000  laths  was  $41.50.     Find  the  rate  of  duty  per  thousand 

(til  cacli. 

18.  If  1  is  added  to  the  numerator  of  a  certain  fraction,  its 
value  becomes  J ;  if  2  is  added  to  the  denominator,  its  value 
becomes  J.     What  is  the  fraction  ? 

Suggestion.  —  F^t  -  =  the  fraction. 

y 

19.  If  each  term  of  a  certain  fraction  is  increased  by  1,  the 
value  of  the  fraction  is  decreased  by  ^^^;  but  if  each  term  is 
decreased  by  1,  the  value  of  the  fraction  is  increased  by  yV. 
What  is  the  fraction  ? 


190  SIMULTANEOUS   SIMPLE   EQUATIONS 

20.  A  certain  numbeu  expressed  by  two  digits  is  equal  to  7 
times  the  sum  of  its  digits ;  if  27  is  subtracted  from  the  num- 
ber, the  difference  will  be  expressed  by  reversing  the  order  of 
the  digits.     What  is  the  number  ? 

Suggestion.  — The  sum  of  x  tens  and  y  units  is  (10a;+  y)  units;  of 
y  tens  and  x  units,  (10  y  -\-x)  units. 

21.  The  sum  of  the  two  digits  of  a  certain  number  is  12, 
and  the  number  is  3  greater  than  6  times  the  sum  of  its  digits. 
What  is  the  number  ? 

22.  When  a  certain  number  expressed  by  two  digits  is 
divided  by  the  sum  of  its  digits,  the  quotient  is  8 ;  and  w^hen 
the  first  digit  is  diminished  by  3  times  the  second,  the  re- 
mainder is  1.     What  is  the  number  ? 

23.  If  a  rectangular  floor  were  2  feet  wider  and  5  feet 
longer,  its  area  would  be  140  square  feet  greater ;  if  it  were 
7  feet  wider  and  10  feet  longer,  its  area  would  be  390  square 
feet  greater.     What  are  its  dimensions  ? 

24.  A  crew  can  row  8  miles  downstream  and  back,  or  12 
miles  downstream  and  halfway  back  in  1^  hours.  What  is  their 
rate  of  rowing  in  still  water  and  the  velocity  of  the  stream  ? 

25.  A  man  rows  12  miles  downstream  and  back  in  11  hours. 
The  current  is  such  that  he  can  row  8  miles  downstream  in 
the  same  time  as  3  miles  upstream.  What  is  his  rate  of  row- 
ing in  still  water,  and  what  is  the  velocity  of  the  stream  ? 

26.  A  train  of  25  cars  loaded  with  iron  ore  was  run  out  on  a 
dock  and  the  ore  emptied  into  pockets  beneath  the  tracks. 
The  ore  filled  7  pockets  and  \  of  another.  To  fill  this  last 
pocket,  then,  required  16  tons  less  than  2  extra  car  loads. 
What  was  the  capacity  of  a  car?   of  a  pocket? 

27.  If  100  pounds  of  soft  coal  in  burning  can  evaporate  50 
pounds  more  water  than  6  gallons  of  oil,  and  if  60  pounds  of 
coal  can  evaporate  10  pounds  less  water  than  4  gallons  of  oil, 
how  many  pounds  of  water  can  1  pound  of  coal  evaporate  ? 
1  gallon  of  oil? 


SIMULTANEOUS  SIMPI.K    KQUATIONS  191 

28.  A  Florida  farmer  shipped  24,300  pineapples  packed  in 
150  crates  of  one  size  and  375  crates  of  another.  Later  he 
shipped  3(J,000  pineapples  in  ()75  crates  of  the  first  size  and 
■)50  crates  of  the  second.  Find  the  capacity  of  a  crate  of  each 
size. 

29.  The  weight  of  a  quantity  of  naphtha  and  petroleum  was 
12,400  pounds.  Each  gallon  of  naphtha  weighed  5J  pounds 
and  cost  GJ^;  each  gallon  of  petroleum  weighed  G.V  pounds  and 
cost  7^^.  If  the  sum  paid  for  the  total  quantity  was  ^145, 
how  many  gallons  were  there  of  each  product  ? 

30.  A  German  dredge  on  trial  removed  in  1^  hours  a  quan- 
tity of  mud  that  at  the  contract  rate  would  have  required  2| 
hours,  removing  1400  cubic  meters  more  per  hour  than  was 
required  by  the  contract.  Find  the  contract  rate  per  hour, 
and  the  actual  rate. 

31.  A  and  B  together  can  do  a  piece  of  work  in  12  days. 
After  A  has  worked  alone  for  5  days,  B  finishes  the  work  in 
26  days.     In  what  time  can  each  alone  do  the  work  ? 

32.  A  quantity  of  wheat  could  be  thrashed  by  two  machines 
in  G  days,  but  the  larger  machine  worked  alone  for  8  days  and 
was  then  replaced  by  the  smaller,  which  finished  in  3  days. 
How  long  would  it  have  taken  the  larger  machine  to  thrash  all 
of  the  wheat  ?  the  smaller  machine  ? 

33.  The  plates  of  a  ship  can  be  riveted  in  30  days  by  10 
gangs  of  riveters,  4  using  hand  hammers  and  6  using  pneumatic 
hammers ;  or  in  32  days  by  10  gangs,  5  of  each  kind.  How 
long  would  it  take  12  gangs  all  using  pneumatic  hammers  ? 

34.  A  and  B  can  do  a  piece  of  work  in  a  days,  or  if  A  works 
711  days  alone,  B  can  finish  the  work  by  working  n  days.  In 
liow  many  days  can  each  do  the  work  ? 

35.  A  can  build  a  wall  in  c  days,  and  B  can  build  it  in  d 
days.  How  many  days  must  each  work  so  that,  after  A  has 
done  a  part  of  the  work,  B  can  take  his  place  and  finish  the 
wall  in  a  days  from  the  time  A  began  ? 


192  SIMULTANEOUS   SIMPLE   EQUATIONS 

36.  At  simple  interest  a  sum  of  money  amounted  to  $2472 
in  9  months  and  to  $2528  in  16  months.  Find  the  amount  of 
money  at  interest  and  the  rate. 

37.  A  man  invested  $4000,  a  part  at  5  %  and  the  rest  at  4  %. 
If  the  annual  income  from  both  investments  was  $175,  what 
was  the  amount  of  each  investment  ? 

38.  A  man  invested  a  dollars,  a  part  at  r%  and  the  rest  at 
s%  yearly.  If  the  annual  income  from  both  investments  was 
h  dollars,  what  was  the  amount  of  each  investment  ? 

39.  A  sum  of  money  at  simple  interest  amounted  to  b 
dollars  in  t  years,  and  to  a  dollars  in  s  years.  What  was  the 
principal,  and  what  was  the  rate  of  interest  ? 

40.  A  certain  number  of  people  charter  an  excursion  boat, 
agreeing  to  share  the  expense  equally.  If  each  pays  a  cents, 
there  will  be  h  cents  lacking  from  the  necessary  amount ;  and 
if  each  pays  c  cents,  d  cents  too  much  will  be  collected.  How 
many  persons  are  there,  and  how  much  should  each  pay  ? 

41.  A  mine  is  emptied  of  water  by  two  pumps  which  together 
discharge  m  gallons  per  hour.  Both  pumps  can  do  the  work  in 
h  hours,  or  the  larger  can  do  it  in  a  hours.  How  many  gallons 
per  hour  does  each  pump  discharge  ?  What  is  the  discharge 
of  each  per  hour  when  a  =  ^,  6  =  4,  and  m  =  1250  ? 

42.  Two  trains  are  scheduled  to  leave  A  and  B,  m  miles 
apart,  at  the  same  time,  and  to  meet  in  h  hours.  If  the  train 
that  leaves  B  is  a  hours  late  and  runs  at  its  customary  rate,  it 
will  meet  the  first  train  in  c  hours.  What  is  the  rate  of  each 
train  ? 

What  is  the  rate  of  each,  if  m  =  800,  c  =  9,  a  =  If ,  and 
6  =  10  ? 

43.  A  man  ordered  a  certain  amount  of  cement  and  received 

it  in  c  barrels  and  d  bags ;  a  barrels  and  h  bags  made  —  of  the 

n 

total  weight.  How  many  barrels  or  how  many  bags  alone 
would  have  been  needed  ?  Find  the  number  of  each,  if  c  =  16, 
d  =  15,  a  =  ^,  b  =  15,  m  =  1,  and  7i  =  2. 


SIMULTANEOUS  SIMPLE   EQUATIONS  193 

THREE  OR  MORE   UNKNOWN  NUMBERS 

255.  Tilt*  student  has  been  solving  systems  of  tico  independ- 
ent siniultaueous  equations  involving  two  unknow  n  numbers. 
In  general, 

Pki-nciple.  —  Every  st/steni  of  independent  simultaneous  simple 
'  (/nations  involving  the  same  number  of  unknown  numbers  as  there 
(ire  equations  can  be  soloed,  and  is  aalisjied  by  one  and  only  one 
set  of  calnes  of  its  unknoicn  nnmhei'S. 

EXERCISES 

rx  +  2y  +  \\z=U,  (1) 

256.  1.    Solve     2xH-2/4-2  2  =  10,  (2) 

. 3x4-4^-32  =  2.  (3) 

Som:tion.— Eliminating  z  by  combining  (1)  and  (3), 

(1)  +  G^),  4a;  +  6y=l(i.  (4) 

Eliminating  z  by  combining  (2)  and  (3), 

(2)  X  3,  6a!+    3y-i-6;2  =  30 
(8)  X  2,  6  ic  +    8  y  -  6  «  =    4 

Adding,  12x+lly  ^U  (6) 

Eliminating  x  by  combining  (5)  and  (4), 
(4)  X  3,  12x  +  18y  =  48  (rt) 

(fi)-(r>),  7y=14;  .-.^  =  2. 

Substituting  the  value  of  y  in  (4),  4  x  +  12  =  10  ;  .-.  at  =  1. 

Substituting  the  values  of  x  and  y  in  ^1), 

1  +  4  +  3  «  =  14  ;  .-.  «  =  3. 

Verification.  —  Substituting  a;  =  1,  y  =  2,  and  ^  =  3  in  the  given 
equations,  ^^ ^  becomes    1  +  4  +  9  =  14,  or  14  =  14 ; 

(2)  becomes    2  +  2  +  fi  =  10,  or  10  =  10 ; 
and  (3)  becomes    3  +  8-0=    2,  or    2=    2; 

that  is,  the  given  equations  are  satisfied  for  y  =  1 ,  y  =  2,  and  z  —  3. 
milne's  stand,  alo.  —  13 


194  SIMULTANEOUS   SIMPLE   EQUATIONS 

Solve,  and  test  all  results : 

'x-\-3y  —  z  =  10,  Ux—  oy-^3z  =  14, 


2. 


9. 


2x  +  oy-{-4:Z  =  :)7, 
3x  —  y-\-2z  =  lo. 


x-{-y  +  z  =  53, 

3.  \x-^2y-{-3z  =  105, 
>*4-32/  +  42  =  134. 

x-yi-z  =  30, 

4.  <3y-x-z  =  12, 
[7z-y  +  2x  =  Ul. 


5.   < 


6.    < 


7. 


Sx-5y-{-2z  =  53, 
x-{-y-z  =  9, 
13x-9y-\-3z  =  71. 

x-^3y-\-4:Z  =  S3, 
x-{-y  +  z  =  29, 
^6x-\-Sy-\-Sz  =  156, 

2x-\-3y  +  4.z  =  29, 
3x  +  2y  +  5z  =  32, 
4.x-^3y-{-2z  =  25. 

3x-2y-\-z  =  2, 
2x-\-5y-{-2z  =  27, 
[x-]-3y-\-3z  =  25. 

2x  —  3y-^4:z  —  v  =  4:, 
4.x-^2y-z-{-2v  =  13, 
x  —  y-\-2z-\-3v  =  17, 
Sx-{-2y-z-^4:V  =  20. 


10. 


11. 


12. 


13. 


14. 


15. 


16. 


\4:X 

X  -\-  7  y  —  z  =  13f 

[2  X  +  5  y  -\-r)z  =  3G. 

2x  +  y-3z-{-4:v:  =  4i, 
3x  —  2y-\-z  —  ir—  —  1, 
4:X  —  y-\-2z-\-tc  =  5o, 
[5x-3y-{-4:Z-w  =  39. 

{7x-l=3y, 
11 2;  -  1  =  7  V, 
42-1  =  7?/, 
19a;-l  =  3v. 

\x  +  iy-\-iz  =  S2, 
ix-\-\y-\-lz  =  lo, 
i^  +  i2/  +  i^  =  12. 


ix-ly-\-iz  =  3, 
\x-ly  +  iz  =  5. 


^±^  +  3z  =  29, 


^x-y  I  2 


99 


3a;-2/  =  3(2J-l). 

3ic  +  ?/  —  ;2  +  2v  =  0, 
32/-2a;H-;z-4iJ=21, 
a;_2/  +  2z-3v  =  6, 
Ux-\-2y-3z-\-v  =  12. 


SIMCLTANEOUS  SIMPLE   EQUATIONS 


195 


17.    Solve  the  equations 


u-\-v-\-x  —  y  =  2y 
u-{-v  —  x-{-y  =  4, 
u-v  +  x-\-y  =  6, 
V  —  w  4-  ic  4-  //  =  8. 

SoLiTios.  —  Adding  the  equations,  2  n  +  2  p  +  2 a;  +  2  y  =  20. 

Dividing  by  2,  u  +  v  +  x  +  y  =  10. 

Subtracting  each  of  the  given  equations  from  this  equation, 
2y  =  8,   2x  =  6,    2t)  =  4,   2m  =  2; 


whence. 


a;  =  3, 


2, 


tt  =  l. 


Solve,  and  test  all  results 

: 

J?4-//  =  9, 

a;  4-32/4-2  =  14, 

18. 

.'/  4-  2;  =  7, 
2  + a;  =  5. 

''  4-  a;  4-  y  =  15, 

22. 

«  4- 2/ 4-32  =  16, 
3a;4.2/  +  2  =  20. 

'2/4-2-hv-a;  =  22, 

19. 

x-\-y  +  z  =  lS, 
3/ -f  2  4- V  =  17, 
2  -f  f  4-  a;  =  16. 

X      y 

23. 

z-|-v4-a;-y=18, 
V  4-  a;  4-  ?/  —  2  =  14, 
.a;  4- 2/ 4- 2  —  ^  =  10. 

[1  +  ^-1  =  0, 
a;     2/ 

20.*^ 

i  +  l  =  8. 

Z        X 

x  +  y     5 

24. 

y     z 
-4---2  =  0. 

Z        X 

xy       1 

x-^y     8' 

21. 

yz  ^1, 

y^z     6 

zx        1 

z4-a?     7 

25. 

.V2     ^^ 

y  +  z     4* 

zar        1 

z  +  x     2 

Srco 

E8T10N.— If   -^^  = 

x  +  y 

1 
=  5' 

zy 

4w 

hence,  -  +  -  =  6. 
y    X 

196 


SIMULTANEOUS   SIMPLE   EQUATIONS 


(1) 

26.    Solve  for  x,  y, 

and  z,  < 

62;j;  —  cxy  +  or?/2;  = 

=  bxyz, 

(2) 

a      ?>      c 
=  c 

(3) 

Solution. 

(1)  +  (3), 

X 

.•.x=    2«. 

a  +  c 

(4) 

(2)  -  xyz, 

y     z     X 

(5) 

(5) -(3), 

Substituti 

nsr  the  value 

s  of  X  and 

w  in  CIV  and  solvi 

n.,.^    2c  . 

Solve  for  x,  y,  z,  and  v : 

-axy-x-y=0, 

27.     < 

bzx  —  z  —  x  =  0, 

cyz  —  y  —  z  =  0. 

iC  +  ?/  —  2  :=  0, 

28.    <x  —  y  =  2  b, 

x-Jt-z  =  3a-\-b. 

'v-\-x  =  2  a, 

29.    . 

x-\-y  =  2a  —  z, 

y-\-z  =  a-^b, 

_v  —  z  =  a-\-c. 

30. 


y  -\-z  —  3  x  =  2  a, 
z-\-x  —  3y  =  2b, 
x  +  y-3z  =  2c, 
2x  +  2y-^v  =  0, 


31. 


32. 


33. 


34. 


abxyz  +  cxy- 

-ayz—bzx=0, 

bcxyz-\-ayz- 

-bzx—cxy  =  0, 

caxyz  +  bzx- 

-cxy—ayz  =  0. 

x-\-y-{-z  = 

a-i-b  +  c, 

x-\-2y  +  3 

z  =  b  +  2c, 

x+3y+4 

z  =  b  +  3c. 

'v-{-x-^y  = 

a-^2b  +  c, 

X-^y-^Z  = 

3  b, 

y+z+v= 

a-hb, 

iZ-\-V-\-  X  = 

a-\-3b-c. 

ax  -^by-\-cz  =  3, 

ct-\-b 
x  +  y  =  — —y 
ab 

y-^^="t 

G^ 

SIMULTANEOUS  SIMPLE   EQUATIONS  197 

Problems 
257.    1.    There  are  three  nuiubers  such  that  the  sum  of  ^ 
ot  the  first,  J  of  the  second,  and  \  of  the  third  is  12 ;  of  ^  of 
the  first,  \  of  the  second,  and  ^  of  the  third  is  9 ;  and  the  sum 
of  the  numbers  is  38.     What  are  the  numbers  ? 

2.  Divide  800  into  three  parts,  such  that  the  sum  of  the  first, 
J  of  the  second,  and  |  of  the  third  shall  be  400 ;  and  the  sum 
of  the  second,  |  of  the  first,  and  \  of  the  third  shall  be  400. 

3.  A  and  B  can  do  a  piece  of  work  in  10  days ;  A  and  C  can 
do  it  in  8  days ;  and  B  and  C  can  do  it  in  12  days.  How  long 
will  it  take  each  to  do  it  alone  ? 

4.  Three  cities,  A,  B,  and  C,  connected  by  straight  roads, 
aro  at  the  vertices  of  a  triangle.  From  A  to  B  by  way  of  C  is 
].;<»  miles;  from  B  to  C  by  way  of  A  is  110  miles;  and  from  C 
to  A  by  way  of  B  is  140  miles.     How  far  apart  are  the  cities  ? 

5.  A  certain  number  is  expressed  by  three  digits  whose  sum 
is  14.  If  693  is  added  to  the  number,  the  digits  will  appear  in 
reverse  order.  If  the  units'  digit  is  equal  to  the  tens'  digit 
increased  by  6,  what  is  the  number  ? 

6.  A,  B,  and  C  have  certain  sums  of  money.  If  A  gives  B 
$100,  they  will  have  the  same  amount;  if  A  gives  C  $100,  C 
will  have  twice  as  much  as  A;  and  if  B  gives  C  $100,  C  will 
have  4  times  as  much  as  B.     What  sum  has  each? 

7.  A  quantity  of  water  sufficient  to  fill  three  jars  of  different 
sizes  will  fill  the  smallest  jar  4  times;  the  largest  jar  twice 
with  4  gallons  to  spare ;  or  the  second  jar  3  times  with  2  gal- 
lons to  spare.     What  is  the  capacity  of  each  jar  ? 

8.  A  contractor  used  4  scows  to  convey  sand  from  his  dredge 
to  the  dumping  ground.  He  was  credited  by  the  inspector  as 
follows : 

Apr.  20,  scows  a,  b,  r,  rf,  a,  h,  c,  d,  a,  and  6,  10  loads,  4230  cu.  yd. 
Apr.  21,  scows  c,  d.  «,  ft,  c,  <l,  a,  6,  c,  and  rf,  10  loads,  4320  cu.  yd. 
Apr.  22,  scows  a,  ft,  r,  d,  a,  ft,  c,  d,  and  a,  9  loads,  3870  cu.  yd. 
Apr.  23,  scows  ft,  c,  d,  a,  ft,  and  c,  6  loads,  2470  cu.  yd. 

Find  the  capacity  of  each  scow. 


GRAPHIC   SOLUTIONS 


SIMPLE  EQUATIONS 

258.  When  related  quantities  in  a  series  are  to  be  compared, 
as  for  instance  the  population  of  a  town  in  successive  years, 
recourse  is  often  had  to  a  method  of  representing  quantities 
by  lines.     This  is  called  the  graphic  metliod. 

By  this  method,  quantity  is  photographed  in  the  process  of 
change.  The  whole  range  of  the  variation  of  a  quantity,  pre- 
sented in  this  vivid  pictorial  way,  is  easily  comprehended  at  a 
glance ;  it  stamps  itself  on  the  memory. 

259.  In  Fig.  1  is  shown  the  population  of  a  town  throughout. 

its  variations  during  the  first 
13  years  of  the  town's  exist- 
ence. 

The  population  at  the  end  of 
2  years,  for  example,  is  repre- 
sented by  the  length  of  the 
heavy  black  line  drawn  upward 
from  2,  and  is  4000 ;  the  popu- 
lation at  the  end  of  6  years  is 
7000;  at  the  end  of  10  years, 
6300  approximately ;  and  so 
Fig.  1.  on. 

260.  Every  point  of  the  curved  line  shown  in  Fig.  1  exhibits 
a  pair  of  corresponding  values  of  two  related  quantities,  years 
and  population.  For  instance,  the  position  of  E  shows  that 
the  population  at  the  end  of  4  years  was  6000. 

Such  a  line  is  called  a  graph. 

198 


" 

r 

- 

f 

» 

/ 

t 

/ 

M 

/ 

/ 

D 

A 

\ 

y 

S3 

A 

4 

- 

•*   E 

-/ 

'b 

1   / 

e 

1    : 

i    4 

. 

>    C 

i 
Ye 

ars 

1 

0  1 

1 1 

I  1 

a  1 

4  1 

5 

GRAHllC    SOLUTlOxXS 


199 


Graphs  are  useful  iu  numberless  ways.  The  statistician  uses  them  to 
l>ivseut  information  in  a  telling  way.  The  broker  or  merchant  uses  them 
to  compai-e  the  rise  and  fall  of  prices.  The  physician  uses  them  to  record 
the  progress  of  diseases.  The  engineer  uses  them  in  testing  materials  and 
ill  computing.  The  scientist  uses  them  in  his  investigations  of  the  laws 
of  nature.  In  short,  graphs  may  be  used  whenever  two  related  quantities 
are  to  be  compared  throughout  a  series  of  values. 

261.  The  graph  in  Fig.  2  represents  the  rate  in  gallons  per 
(lay  per  person  at  which  water  was  used  in  Kew  York  City 
diu-iug  a  certain  day  of  24  hours. 


110  - 

/ 

r 

\ 

/ 

\ 

/ 

\, 

r 

>- 

i.1. 

i 

/ 

V 

W) 

I 

/ 

._^ 

no  - 

I 

" 

"■ 

— 

~ 

~ 

\R 

s 

i 

s 

» 

n 

>ai 

vf 

[?" 

1^ 

[id 

Cjg 

lit 

0       I    2 
MidulKht 


5   0 


12 


Fig.  2. 


18  -.'l  24 

6  P.M.  MIduight 


Thus,  if  each  horizontal  space  represents  1  hour  (from  mid- 
night) and  each  vertical  space  10  gallons,  at  midnight  water 
was  being  used  at  the  rate  of  about  84  gallons  per  day  per  per- 
son; at  G  A.M.,  about  91  gallons;  at  1  p.m.,  the  13th  hour, 
about  108  gallons;  etc. 

1.  What  was  the  approximate  consumption  of  water  at 
2  A.M.  ?  at  noon?  at  1  :  30  p.m.  ?  at  2  :  30  p.m.  ?  at  6  p.m.  ? 

2.  "What  was  the  maximum  rate  during  the  day  ?  the  mini- 
mum rate?  at  what  time  did  each  occur? 

3.  During  what  hours  was  the  rate  most  uniform  ?  What 
was  the  rate  at  the  middle  of  each  hour  ? 

4.  What  was  the  average  increase  per  hour  between  6  a.m. 
and  8  a.m.?   the  average  decrease  between  4  p.m.  and  8  p.m.? 


200 


GRAPHIC   SOLUTIONS 


262.  Fig.  3  gives  a  part  of  the  graph  that  shows  the  rela- 
tion between  numbers  and  their  respective 
squares,  horizontal  distances  representing 
the  numbers  1,  2,  3,  etc.,  and  vertical  dis- 
tances, their  squares.  From  this  graph  we 
may  read  the  squares  or  the  square  roots 
of  various  numbers. 

Thus,  the  square  of  3  is  represented  by  the 
vertical  line  extending  from  3  to  the  graph. 
It  is  9.  Conversely,  the  squai-e  root  of  9  is 
represented  by  the  horizontal  line  that  ex- 
tends from  9  to  the  graph.     It  is  3. 

Similarly,  the  square  of  21  is  represented 
by  a  vertical  line  halfway  between  2  and  3 
and  extending  to  the  graph.  It  is  6i  The 
square  root  of  3  is  represented  by  a  horizon- 
tal line  from  3  to  the  graph.  It  is  approxi- 
mately 1.7. 

Eind  from  the  graph  the  square  of  4; 
of  1^ ;  of  3 J ;  the  approximate  square  root 
of  11 ;  of  13  ;  of  8 ;  of  2\. 


27 

26 
25 
24 
23 
22 
21 
20 
10 
18 
17 
16 

— 

— 

— 

— 

— 

, 

14 
13 
12 
11 

/ 

/ 

/ 

/ 

/ 

8 

' 

._ 

._. 

4 

fl 

1 

1 

/ 

j 

1 

/ 

/  ! 

12    3    4    5    6 

Fig.  3. 


263.  Let  X  and  y  be  two  algebraic  quantities  so  related  that 
2/  =  2  ic  —  3.  It  is  evident  that  we  may  give  x  a  series  of 
values,  and  obtain  a  corresponding  series  of  values  of  y ;  and 
that  the  number  of  suck  pairs  of 
values  of  x  and  y  is  unlimited.  All 
of  these  values  are  represented  in 
the  graph  of  y  =  2  x  —  3.  Just  as 
in  the  preceding  illustrations,  so  in 
the  graph  of  y=2x  —  3,  Fig.  4, 
values  of  x  are  represented  by  lines 
laid  off  on  or  parallel  to  an  x-axis, 
X'X,  and  values  of  y  by  lines  laid 
off  on  or  parallel  to  a  y-axis,  Y'Y, 
usually  drawn  perpendicular  to  the 


ic-axis. 


Fig.  4. 


GRAPHIC   SOLUTIONS 


201 


For  example,  the  position  of  P  shows  that  y  =  3  when  a;  =  3 ; 
the  position  of  Q  sliows  that  y  =  o  when  a;=  4 ;  the  position  of 
H  shows  that  y  —  l  when  x  —  iy\  etc. 

Evidently  every  point  of  the  graph  gives  a  pair  of  corre- 
sponding values  of  x  and  y. 

264.  Conversely,  to  locate  any  point  with  reference  to  two 
axes  for  the  purpose  of  representing  a  pair  of  corresponding 
values  of  x  and  y,  the  value  of  x  may  be  laid  off  on  the  a?-axis 
as  an  x-di stance,  or  abscissa,  and  that  of  y  on  the  y-axis  as  a 
;i-distance,  or  ordinate.  If  from  each  of  the  points  on  the  axes 
obtained  by  these  measurements,  a  line  parallel  to  the  other 
axis  is  drawn,  the  intersection  of  these  two  lines  locates  the 
point. 

Thus,  in  Fig.  4.  to  represent  the  corresponding  vahies  re  =  3,  y  =  3,  a 
point  P  may  he  located  by  measuring  3  units  from  O  tu  M  on  the  x-axis 
and  3  units  from  O  to  N  on  the  y-axis,  and  then  drawing  a  line  from  M 
parallel  to  OF,  and  one  from  .V  parallel  to  OX, producing  these  lines 
until  they  intersect. 

265.  The  abscissa  and  ordinate  of  a  point  referred  to  two 
perpendicular  axes  are  called  the  rectangular  coordinates,  or 
simply  the  coordinates,  of  the  point. 

Thus,  in  Fig.  4,  the  coordinates  of  P  are  0M{=  NP)  and  MP{  =  ON). 

266.  I'y  universal  custom  positive  values  of  x  are  laid  off 
from  0  as  a  zero-point,  or  origin,  toward  the  rights  and  neg- 
ative values  toward  the  left.     Also 

positive  values  of  y  are  laid  off  w^> 
irurd  and  negative  values  doivnward. 

The  point  A  in  Fig.  5  may  be 
designated  as  "  the  point  (2,  3),"  or 
by  the  equation  A  =  (2,  3). 

Similai'ly, 
^=(-2,  4),  C=(-3,   -1),  and 

n     ,1.  -2). 

The  abscissa  is  always  written  first. 


-] 

■^ 

Y 

~ 

~^ 

~" 

"~ 

B 

'a 

x' 

X 

3- 

>- 

0 

1 

2 

3 

4 

d 

-1 

'd 

Y' 

202 


GRAPHIC   SOLUTIONS 


267.    Plotting  points  and  constructing  graphs. 


EXERCISES 

Note,  —  The  use  of  paper  ruled  iu  small  squares,  called  coordinate 
paper,  is  advised  in  plotting  graphs. 

Draw  two  axes  at  right  angles  to  each  other  and  locate : 

1.  A  =  (3,  2).  5.  E=  (5,  5).  9.  i  =  (0,  4). 

2.  j8  =  (3,  -2).  6.  i^=(-5,  5).  10.  iJf=(0,  -r>). 

3.  0=(4,  3).  7.  G  =  (-2,5).  11.  .¥=(3,0). 

4.  Z)  =  (4,  -3).  8.  //=(-3,  -4).  12.  P  =  (-6,0). 

13.  Where  do  all  points  having  the  abscissa  0  lie  ?  the 
ordinate  0? 

14.  What  are  the  coordinates  of  the  origin  ? 

15.  Construct  the  graph  of  the  equation  2y  —  x  =  2. 

Solution 

Solving  for  y,  y  =  ^{x  +  2). 

Values  are  now  given  to  x  and  computed  for  y  by  means  of  this 
equation.  The  numbers  substituted  for  x  need  not  be  large.  Con- 
venient numbers  to  be  substituted  for  x  in  this  instance  are  the  even 
integers  from  —  6  to  +  6. 

When  x  =  —  6,y  =  —  2.     These  values  locate  the  point  ^  =  (  —  6,  —  2). 

When  ic  =  —  4, 2/  =  —  1.     These  values  serve  to  locate  5=(—  4,  — 1). 

Other  points  may  be  located  in  the  same  way. 

A  record  of  the  work  should  be  kept  as  follows : 

2/  =  K^  +  2) 


— 

~ 

Y 

~ 

~ 

X 

- 

- 

A 

^ 

>" 

fe 

- 

'l^ 

y^ 

r 

^ 

X* 

^ 

X' 

U 

^ 

D 

X 

^ 

'^ 

f^ 

B 

r 

^ 

[A 

Y' 

^ 

_ 

_ 

Fig.  6. 


X 

// 

I'OINT 

-6 

-2 

.4 

-4 

-1 

^ 

-2 

0 

c 

0 

1 

B 

2 

2 

E 

4 

.T 

F 

6 

' 

a 

A  line  drawn  through  A,  B,  C,  2>,  etc.,  is  the  graph  of  2 


GRAPHIC   SOLUTIONS  203- 

Constnict  the  graph  of  each  of  the  following: 

16.  !/=:3x  —  7,  19.   3  a;  — ?/  =  4.  22.  'Sx  =  2ij. 

17.  y  =  2x+i,  20.   4.r- //=!().  23.  2  a; +  ?/  =  !. 

18.  y  =  2x—i.  21.    x-'J>/  =  'J.  24.  2x-{-Sy  =  (j. 

268.  It  can  be  proved  by  the  principle  of  the  similarity  of 
trianj^les  that : 

Principle. —  The  graph  of  a  simple  equation  is  a  straight  line. 
For  this  reiison  simple  equations  are  sometimes  called  linear 
equations. 

269.  Since  a  straight  line  is  determined  by  two  points,  to 
plot  the  graph  of  a  linear  equation,  p/nf  fu:o  points  and  draw 
a  straight  line  through  them. 

It  is  often  convenient  to  plot  the  points  where  the  graph 
intersects  the  axes.  To  find  where  it  intersects  the  a-axis,  let 
7  =  0;  to  find  where  it  intersects  the  y-axis,  let  x  =  0. 

Thus,  in  y  =  i  (a;  +  2),  when  y  =  0,  a;  =  —  2,  locating  C,  Fig.  6  ;  when 
a:  =  0,  y  =  1,  locating  D. 

Draw  a  straight  line  through  C  and  Z>. 

If  the  equation  has  no  absolute  term,  a;  =  0  when  y  =  0,  and  this 
method  gives  only  one  point.  In  any  case  it  is  desirable,  for  the  sake  of 
accuraq/,  to  plot  points  some  distance  apart,  as  A  and  Gj  in  Fig.  6. 

EXERCISES 

270.  Construct  the  graph  of  each  of  the  following: 

1.  y  =  x-2.  8.  2x-3y  =  C).  15.    Sx-Sy=-6. 

2.  y  =  2-x.  9.  3x4-4^  =  12.  16.    -2x  +  y=-3. 

3.  v  =  9  — 4a;.  10.  ox  — 2 y  =  10.  17.    —  3a;  +  4y=8. 

4.  y  =  ^x-9.  11.  7a;-y  =  14.  18.   5x  +  Sy  =  7^. 

5.  y  =  10  — 2  a;.  12.  4  — x  =  2»/.  19.   x  —  ^y  =  3. 

6.  t/  =  2a;-10.  13.  2a;4-3y  =  0.  20.   ix  +  iy  =  2. 

7.  2/=-2a;-4.  14.  x-^y-3  =  0.  21.    .7a;-.3y  =  .4. 


•  204 


GRAPHIC   SOLUTIONS 


n 

n 

- 

\ 

/ 

d 

\ 

iX 

y 

D 

\ 

V 

p^ 

w 

\ 

'^ 

N 

/ 

\ 

H 

/ 

\ 

•R 

/ 

^ 

f- 

/ 

\ 

/ 

A 

0 

3  M 

\ 

/ 

\ 

/ 

Fig.  7. 


271.   Graphic  solution  of  simultaneous  linear  equations. 

1.  Let  it  be  required  to  solve  graphically  the  equations 

(y  =  2^x,  (1) 

\y  =  Q~x.  (2) 

As  in  §  267,  construct  the 
graph  of  each  equation,  shown 
in  Fig.  7. 

1.  When  a;  =  —  1,  the  value 
of  y  in  (1)  is  represented  by 
AB,  and  in  (2)  by  AG. 

Therefore,  when  a;  =  —  1, 
the  equations  are  not  satis- 
fied by  the  same  values  of  y. 

2.  Compare  the  values  of  y  when  a?  =  0  ;  when  x  =  1 ;  2. 

3.  For  what  value  of  x  are  the  values  of  y  in  the  two  equa- 
tions equal,  or  coincident  ? 

4.  What  values  of  x  and  y  will  satisfy  both  equations  ? 
The  required  values  of  x  and  y,  then,  are  represented  graphi- 
cally by  the  coordinates  of  P,  the  intersection  of  the  graphs. 

II.    Let  the  given  equations 

be  I       ^  +  2/  =  7, 

5.  What  happens  if  we  try 
to  eliminate  either  x  ov  y? 

6.  Since  y  =  7—x  in  both 
equations,  what  will  be  the 
relative  positions  of  any  two 
points  plotted  for  the  same 
value  of  X?  the  relative  posi- 
tions of  the  two  graphs  ? 

7.  The   algebraic    analysis    shows   that   the   equations   are 
indeterminate. 

The  graphic  analysis  also  shows  that  the  equations  are  inde- 
terminate, for  their  grajjhs  coincide. 


s 

s 

s . 

4^^ 

^c 

^^ 

3SS 

^^ 

3Sc- 

^ 

^ 

Fig.  8. 


GRAPHIC   SOLUTIONS 


205 


be 


^"y     t: 

f 

^c-^ 

5  s 

s  s 

5  ^^ 

s^ 

SrS' 

$^^ 

?           ^^S^_     _x 

_ 

S       ^ 

^ 

y' 

Fia,  9. 


III.   Let  the  given  equations 
p  =  6-x,  (1) 

h,  =  A-x.  (2) 

8.  When  x=—l,  how  much 
greater  is  the  value  of  y  in  (1) 
than  in  (2),  as  shown  both  by 
the  equations  and  their  graphs  ? 

9.  Compare  the  y's  for  other 
values  of  x. 

10.  For  every  value  of  x  the  values  of  y  in  the  two  equa- 
tions differ  by  2,  and  the  graphs  are  2  units  apart,  vertically. 

In  algebraic  language,  the  equations  cannot  be  simultaneous ; 
that  is,  they  are  inconsistent. 

In  graphical  language,  their  gra2)hs  cannot  intersect^  being 
parallel  straight  lines. 

272.  Principles.  —  1,  A  single  linear  equation  involving 
two  unknown  numbers  is  indeterminate. 

2.  Two  linear  equations  involving  two  unknown  numbers 
are  detei'mina^e,  provided  the  equations  are  independent  and 
simultaneous. 

They  are  satisfied  by  one,  and  only  one,  pair  of  common  values. 

3.  The  pair  of  common  values  is  represented  graphically  by 
the  coordinates  of  the  intersection  of  their  graphs. 

EXERCISES 


T 

/. 

V, 

S.^ 

"=^. 

^^ 

^s,^ 

^w 

^^ 

/^ 

M 

b^l 

l^^^. 

X'  .^^_:3 

l--^r__x 

7            t 

y            —\ 

Y' 

:: 

273.    1.    Solve  graphically  the 

6, 
12. 


equations 


[\y-?.x 
X^x^'^y 


Solution.  —  On  plotting  the  graphs 
of  both  equations,  as  in  §  207,  it  is 
found  that  they  intersect  at  a  point 
P,  whose  coordinates  are  1.8  and  2.8, 
approximately. 

Hence,     x  =  1.8  and  y  =  2.8. 


Fig.  10. 
The  cuordiuate^  of  /'are  estimated  to  the  nearest  tenth. 


206 


GRAPHIC   SOLUTIONS 


Note.  —  In  solving  simultaneous  equations  by  the  graphic  method  the 
same  axes  must  be  used  for  the  graphs  of  botli  equations. 

Construct  the  graphs  of  each  of   the  following  systems  of 
equations.    Solve,  if  possible.    If  there  is  no  solution,  tell  why. 


2. 


3. 


6. 


7. 


x-\-y  =  9. 

(x-{-y  =  S, 
[x-\-2y  =  4t. 

x  =  4:-\-y, 
y  =  S-i-x. 

2x-y  =  o, 
4  a? +  2/ =  16. 

8x  =  y-j-9, 


(3x  = 

[2y  = 


y  =  6x-lS. 

y  =  4:x, 
x-y  =  8. 


[y  = 


x  =  i(y  +  4:), 
2(x  -  2). 


9. 


10. 


11. 


12. 


'x-\-y=  -3, 
x-2y=-12. 

y  =  2-x. 

21  =  2(2x  +  y). 

a; +  2/ =  8, 
2x-6y=-9. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


2  x  —  5y  =  5, 
10y  =  2x-\-l. 

(3y  =  2x-7, 
\2x  =  6-}-3y. 

(S(x-4)  =  2y, 
'6(y-\-6)  =  9x. 

10  a;  4-?/ =  14, 
8x-oy=  -2. 

2x-\-Sy  =  S, 
Sx  +  2y  =  S. 

f  4  2/  +  3  £c  =  5, 
{4:X-3y  =  S. 

x-\-3y=-6, 
2x-4:y=-12. 

r4:X-10y  =  0, 

[2x  +  y  =  12. 


21.     ^ 


(x-2y  =  2, 
[2y-6x  =  3. 


22.     <{ 


(3x  +  4.y  =  10, 

[6x  +  8y  =  20. 

fa^  +  |^  =  3i 
10x-2y  =  14^ 


INVOLUTION 


274.  The  process  of  finding  any  required  power  of  an  ex- 
])  less  ion  is  called  involution. 

275.  By  the  definition  of  a  power,  when  «  is  a  positive 
integer  a"  means  a  •  a  •  a  •  ••  to  n  factors. 

The  following  illustrate  powers  of  positive  numbers,  of  nega- 
tive numbers,  of  powers,  of  products,  and  of  quotients,  and 
show  that  every  case  of  involution  is  an  example  of  multipli- 
cation of  equal  factors. 


POWERS  OF  A 

POWERS    OF    A 

POWERS   OF   A 

»SIT1V£    NUMBER 

NEGATIVE    NUMBER 

POWER 

2  =  21 

-2  =  (-2)' 

4  =  2^ 

2 

4  =  2'* 

_2 

4  =  (-2)' 

4 

2 
8  =  2« 

-2 

-8  =  (-2)' 

4 

64  =  (2^»  =  2« 

2 

1G  =  1>^ 

-2 

16  =  (-2)' 

4 

256=(2^*  =  2« 

POWER   OF   A    PRODUCT 

(2 . 3)«  =  (2  •  3)  X  (2 .  3)  =  2  . 2  .  3  . 3  =  2-' .  3*. 

POWER   OF    A    QUOTIENT 

/2Y^2  2^2* 
VV      3*3     3** 

The  last  two  examples  illustrate  the  distributive  law  for 
involution. 


208  JXVOLrTIOX 

276.    Principles.  —  1.    Law   of   Signs   for   Involution. — All 

powers  of  a  positioe  mmiher  are  positive  ;  even  powers  of  a  nega- 
tive number  are  positive,  and  odd  powers  are  negative. 

2.  Law  of  Exponents  for  Involution.  —  The  exponent  of  a  ijower 
of  a  7iumber  is  equal  to  the  exponent  of  the  number  multiplied  by 
the  exponent  of  the  power  to  ivhich  the  number  is  to  be  raised. 

3.  Distributive  Law  for  Involution.  —  Any  jwwer  of  a  product 
is  equal  to  the  product  of  its  factors  each  raised  to  that  power. 

Any  power  of  the  quotient  of  two  numbers  is  equal  to  the  quo- 
tient of  the  ^lumbers  each  raised  to  that  power. 

The  above  laws  may  be  established  for  positive  integral  expo- 
nents as  follows : 

Let  m  and  n  be  positive  integers. 

1.  Principle  1  follows  from  the  law  of  signs  for  multiplication. 

2.  By  notation,  §  27,      (a'")'*  =  a"*  •  a"' .  a*"  •••  to  n  factors 

S  38  =  (i^+»^+f*+—  to  n  terms 


By  notation. 

_  ^m«^ 

3.    By  notation. 

(aft)"  -abx  abx  ab--- to  n  factors 

§82, 

=  (aaa  ••■)(bbb  •••)  each  to  n  factors 

By  notation, 

=  a»b". 

Also 

f«V  =  ^x^x^...  tor.  factors 
\bJ        b      b      b 

§207, 

aaa  •••to  n  factors 
bbb  •  •  •  to  n  factors 

By  notation, 

~b^' 

277.  Axiom  6.  —  TJie  same  powers  of  equal  numbers  are  equal. 
Thus,  if  X  =  3,  a;2  =  3^,  or  9  ;  also  x^  =  3*,  or  81 ;  etc. 

278.  Involution  of  monomials. 

EXERCISES 

1.    What  is  the  third  power  of  4  a^b  ? 
Solution.     (4  a^by  =  4  a^b  x  4  a^^  x  4  a"^6  =  64  a^b^. 


INVOLUTION  2Ui^ 

2.    What  is  the  fifth  power  of  -  2  ab^  ? 

Soi.r  riov 

(  _  2  „!,: ,..      -  -J  ,//,-  .       -J  <,/;-  <  ^-  -J  .fh-  X  -    2  <(h-  X  -  2  a^-  =  -  32  a«6»o. 

To  raise  an  integral  term  to  any  power: 

KuLK. — Raise  the  numerical  coefficient  to  the  required  poicer 
and  annex  to  it  each  letter  tcith  an  exponent  equal  to  the  product 
of  its  exptment  by  the  exjx)tient  of  the  required  potver. 

Make  the  j)oiver  positive  or  negative  according  to  the  law  of 
signs. 

Raise  to  the  power  indicated : 


3 

(a6V)2. 

12. 

i-^c^y- 

21. 

(- 

-X)". 

4. 

(a'b'cy. 

13. 

(-2aVV. 

22. 

(- 

- 1)'^« 

5 

(2  a'cf. 

14. 

(abcx)"*. 

23. 

(- 

-1)-. 

6. 

(7  ahn")' 

15. 

(2eV)«. 

24. 

(- 

-  b)"'^'. 

7 

i-\y. 

16. 

(3  6c)^ 

25. 

(- 

-  6^)2«+>. 

8. 

(-aby. 

17. 

(2  aV)-. 

26. 

(- 

-  a'fj^c'^-^dy. 

9. 

(-3c)». 

18. 

(-2l*m'dy. 

27. 

(- 

-  ar'^f^z'y. 

10. 

{-10  ^y. 

19. 

i-a'x^r-r- 

28. 

(- 

-a"-'6"-=*c)». 

11 

(-6a^3^\ 

20. 

i-x^i/z^-y. 

29. 

[- 

-2  {a -by  J. 

30. 

What  is  the 

square  of       ^  ^,  J 

SOUTION 

\      1  b-^cj  7  h^c  7  b^c      41)  b^d^  ' 

To  raise  a  fraction  to  any  power : 

Rule.  —  Raise  both   numerator  and   denominator  to  the  re- 
quired power  and  prefix  the  proper  si'/,,   to  the  result. 

mii.vk'-  «tv\i>.    ai.«;. — 14 


210  INVOLUTION 

Raise  to  the  power  indicated  : 


.».  I'J^Y-       .T.  (^-ly.      «.  c-?'"^' 


{-!-:)•- 


i  yi  \     6x1  \     a-x 


3-lY.  38.     (-^.  43.     ('^ 


34.  fm\       39.  f-^y.      44  ^«"-'^ 


3?//  \^     ic^^/y  \x-'y 


35.     (;7^^)-  40.     (^^T  45 


2?>"-^ 


«- 


279.  Involution  of  polynomials. 

The  following  are  type  forms  of  squares  of  polynomials  : 
§  105,  (a  +  xf  =a'-^2ax-\-  x-. 

§108,  (a-xy=a^-2ax  +  x'. 

§  111,         (a-x  +  yy-=  a-  +  x'  -\-y~ -2  ax-{-2  ay -2 xy. 

EXERCISES 

280.  Raise  to  the  second  power: 

1.    2a-{-b.  5.  3x—4f.  9.  a  —  b  +  x~y. 

2.2a  —  b.  6.  om"  — 11.  10.  a''' -^  x"  —  y''+\ 

3.  a'"-3  6".  7.  l-3a?>c.  11.  2a-h3?>-4c. 

4.  a--2arr  8.  4d;4  +  5.  12.  oa^ -l+47rl 

Raise  to  the  required  power  by  multiplication : 

13.  (x-\-yy.  15.    (x  +  ijy.  17.   (x-^yy. 

14.  (x  —  y)^  16.    (;v— ?/)^  18.    (.i*  —  ?/)^ 


INVOLUTION  211 

281.    Involution  of  binomials  by  the  Binomial  Theorem  (§  549). 
By  actual  uiultiplication, 

(a  4- «)•' =  a* -f  3  a'a;  +  3aiB* -f- a^. 

(a  +  x}*  =  (i*-\-4  a^x  -f-  (>  a V  4.  4  oar'  +  «*. 

{a  —  xy  =  a*  —  4  a^x  +  6  rr'ar^  —  4  aa:'  +  a;^. 

(a  +  i**)''  =  ^«''  +  5  a*ar  +  10  aV  +  10  aV  -h  r>  aa;*  -f-  ar*. 

(a  -a:/  =  a'^ -  5 a^a; -|-  10  aV  -  10 aV  +  5  aa;* -ar*. 

From  the  expansions  just  given  the  following  observations 
may  be  made  in  regard  to  any  jmsitive  integral  power  of  any 
binomial,  a  standing  for  the  first  term  and  x  for  the  second: 

1.  TTie  number  of  terms  is  one  greater  than  the  index  of  the 
required  ]X)wer. 

2.  Tlie  Jirst  term  contains  a  onhi :  the  last  term  x  only;  all 
other  terms  contain  both  a  and  x. 

3.  The  exponent  of  a  in  the  Jirst  term  is  the  same  as  the  index 
of  the  required  jKtwer  and  it  decreases  1  in  each  succeeding  term; 
the  exponent  of  x  in  the  second  term  is  1,  and  it  increases  1  in 
ea>ch  succeeding  term. 

4.  In  each  term  the  srnn  of  the  exponents  of  a  and  x  is  equal 
to  the  index  of  the  required  power. 

/>.  The  coefficient  of  the  first  term  is  1 ;  the  coefficient  of  the 
second  term  is  the  sam^e  as  the  index  of  the  required  power. 

6.  The  coefficient  of  any  term  may  be  found  by  multiplying 
the  coefficient  of  the  preceding  term  by  the  exponent  of  a  in  that 
term,  and  dividing  this  product  by  ihe  number  of  the  term. 

7.  All  the  terms  are  positive,  if  both  terms  of  the  binomial  are 
lH}sitive. 

8.  The  terms  are  alternately  positive  awl  negative,  if  the  secwid 
term  of  the  binomial  is  negative. 


212  INVOLUTION 

EXERCISES 

282.    1.   Find   the   fifth  power  of  (b  —  y)  by  the  binomial 
theorem. 

Solution 

Letters  and  exponents,  h^         h^y  h^y^  b'^y^         by^      y^ 

Coefficients,  1  5  10  10  6  1 

^igns, +         -  + - +  - 

Combined,  6^  _  5  54^  +  10  53^2  _  10  bY'  +  ^by^  -  y^ 

Expand : 

2.  (m  +  w)-'.  10.  {x  —  yy.  18.  (x  +  4)^ 

3.  (m-?iy.  11.  (c-vy.  19.  (x  +  oy. 

4.  (a  —  cy.  12.  (x  —  ay.  20.  (x  —  2y, 

5.  (a  +  by.  13.  (d-yy.  21.  (a-\-bcy. 

6.  (b-\-dy.  14.  (b  +  yy.  22.  (ab  -  cy. 

7.  (g-ryi  15.  {m-\-ny.  23.  (w-iJii)^ 

8.  (c  +  dy.  16.  (x  +  2y.  24.  (ax-byy. 

9.  (a; 4-^)'.  17.  («  +  3)l  25.  (ax-byy. 

26.  Expand  (a  —  o;)^;  then  (2  r  —  oy  by  the  same  method. 

Solutions 
(«  _  5c)4  =  a4  -  4  a^x  +  6  a^:^^  _  4  ^^^3  +  x*. 
(2  c2  -  5)4  =  (2  c2)4  _  4  (2  c2)3  5  +  6  (2  0^)2  5'^  -  4  (2  c-)6^  +  5* 
=  16  c«  -  160  c6  +  600  c4  -  1000  c^  +  625. 

27.  Expand  (1  +  ^')'^- 

Solution 

(1  -f  a:2)3  =  13  +  3  (l)2(a;-2)  +  3(l)(r;2)2  +  (a;2)3 

Test.  — When  x  =  1,  (1  +  x^y  =  8,  and  I  +  Sx'^ +  Sx^  +  x^  =  S  ;  hence, 
(1  +  x2)3  =  1+3x2  +  3x*  +  X®,  and  the  expansion  is  correct. 


INVOLUTION 

21 

Expand,  and  test  results  : 

28.    {x-\-2y)\ 

32.    (l-3a:»)^ 

36. 

(1-^/. 

29.    {2x-yf. 

33.    (.")/--/(//)•■. 

37. 

(i-2xy. 

30.    C2x-5f, 

34.    {\-ha'b-)\ 

38. 

(x-if. 

31.    (x-'-lO)*. 

35.    (2fea;-6)^ 

39. 

(i^-iyy- 

Expand : 

» {="+1)' 

-  (»<■•-!)• 

46. 

ih-^'i 

"■  e-i)' 

"^  ('-¥)•• 

47. 

e-y- 

-  e-ij- 

-  (i-¥)' 

48. 

(-3' 

49.  Kxpand  {a  —  b  —  f)"\ 

Solution 

(«  — 6-c)'*=(a  — 6  — c)*,  a  binomial  form 

=  (a_6)«-3(a-6y^(!  +  3(a-6)c2-c« 

=  a»  -  3  a2ft  +  .^  ah-^-  b^-3  c(rt2-  2  a6  +  fe2)  +  3  ac«-3  ftc^  -  c» 

=a«-8a2ft+3  a62_  63_3  ^ac+o  a6c-3  ft^c+S  ac2-3  ftc^-c*. 

50.  Expand  (a-\-b  —  c-  d)\ 

SuocESTiox.     (a  +  ^  —  c  —  d)"  =  (a  +  6  —  c  +  d)*,  a  binomial  form. 

Expand : 

51.  (a-\-x-y)\  57.    (a+2b-3cy. 

52.  (a  — m-n)».  58.    («  +  6  +  a;  +  y)*. 

53.  {a  —  x-\-yf.  59.    («-f- 6-aJ  — y)*. 

54.  (a  —  x  —  yf,  60.    (a  —  ?>  +  a;  -  2/)l 

55.  (a  +  a; 4- 2f.  61.    (a-b-x-{-yy. 

56.  (a  — X  — 2)*.  62.    {a  —  b  —  x  —  yf. 
The  Binomial  Theorem  will  be  treated  more  fully  in  §§  649-667. 


EVOLUTION 


283.  The  process  5-  =  5  •  5  =  25  illustrates  involution. 
The  process    V25  =  V5  '5  =  5   illustrates  evolution,  which 

will  be  defined  here  as  the  process  of  finding  a  root  of  a  num- 
ber, or  as  the  inverse  of  involution. 

For  example,  V25  =  5,  for  5-  =  25 ; 

^^^8  =  - 2,  for  (-  2)«=  -  8. 

In  general,  the  nth  root  of  a  is  a  number  of  which  the  nth  power 
is  a. 

284.  Since  25  =  5^  and  also  25  =  ( -  5)  ( -  5)  =  (  -  5)^, 

V25  =  +  5  or  -  5. 

The  roots  may  be  written  together  thus:  ±  5,  read  'j^his  or 
minus  five.' 

Or  they  may  be  written  T  5,  read  '  minus  or  2:)lus  Jive.'' 

Similarly,  V36==  ±  6,  V49  =  ±  7,  V|  =  ±  |. 
Every  positive  yiumher  has  two  square  roots. 

285.  The  square  root  of  —16  is  not  4,  for  4- =  +  16;  nor 
—  4,  for  (—4)^=4-16.  No  number  so  far  included  in  our 
number  system  can  be  a  square  root  of  — 16  or  of  any  other 
negative  number. 

It  would  be  inconvenient  and  confusing  to  regard  Va  as  a 
number  only  when  a  is  positive.  In  order  to  preserve  the 
generality  of  the  discussion  of  number,  it  is  necessary,  there- 
fore, to  admit  square  roots  of  negative  numbers  into  our  num- 
ber system.     The  square  roots  of  — 16  are  written 


V- 16  and  -V-16. 
214 


EVOLUTION  215 

Such  numbers  are  called  imaginary  numbers  and,  in  contrast, 
numbers  that  do  not  involve  a  square  root  of  a  negative  num- 
ber are  called  real  numbers. 

Having  extended  the  number  system,  we  may  now  state  the 
principle  that  ecenj  number  has  two  square  roots,  one  positive 
and  the  other  negative. 

286.  Just  as  every  number  has  two  square  roots,  so  every 
number  has  three  cube  roots,  four  fourth  roots,  etc. 

For  example,  the  cube  roots  of  8  are  the  roots  of  the  equa- 
tion xi^  =  S,  which  later  will  be  found  to  be 

2,  -  1  +  V-^,  and  - 1  -  V-  ;i 

The  present  discussion  is  concerned  only  with  real  roots. 

287.  Since  2-'' =  8,  ^  \/8=2. 
Since      (-2/= -8,                                  V^  =  -2. 
Since            2^  =16  and  (-2)^  =  16,           ^/T6  =  ±2. 
Since            2*  =  32,                                     </82  =  2. 
Since     (-2)'=-.".2,                              </332  =  -2. 

A  root  is  odd  or  even  according  as  its  index  is  odd  or  even. 
It  follows  from  the  law  of  signs  for  involution  that : 

Law  of  Signs  for  Real  Roots. — An  odd  root  of  a  mimber  has 
the  same  sign  as  the  number. 

An  even  root  of  a  positive  number  inay  have  either  sUjn. 
An  even  root  of  a  negative  number  is  imaginary. 

288.  A  real  root  of  *a  number,  if  it  has  the  same  sign  as  the 
number  itself,  is  called  a  principal  root  of  the  number. 

The  principal  square  root  of  2")  Ih  o,  but  not  —  5.  The  i»rincipal  cube 
root  of  8  is  2  ;  of  —  8  is  —  2. 

289.  Axiom  7. —  Hie  same  roots  of  equal  numbers  are  equal. 

Thus,  if  X  =  16,  v'x  =  4  ;  if  x  =  8,  \/x  =  2  ;  etc. 


216  EVOLUTION 

290.  Since  (22)"  =  22^3  =  2\  the  principal  cube  root  of  2«  is 

■v/2^  =  2^  ^3  =  21 

Law  of  Exponents  for  Evolution.  —  Tlie  exponent  of  any  root 
of  a  number  is  equal  to  the  exponent  of  the  number  divided  by 
the  index  of  the  root. 

291.  1.  Since  (5  ay  —  5-a'  =  25  «-,  the  principal  square  root 
of  25  a^  is 


V25a-'^=  V25-  Va^-  =  5a. 


\ 


/a\  ^     a*  a* 

2.    Since  (  -  ]  =  - ,  the  principal  fourth  root  of  —  is 
V6/       6^        _  b* 

J  a*  _  -y/a'^  _  a 

'    h'~  ^^~b' 

Distributive  Law  for  Evolution.  —  Any  root  of  a  2iroduct  may 

be  obtained  by  taking  the  root  of  each  of  the  factors  and  finding 

the  product  of  the  residts. 

Any  root  of  the  quotient  of  two  mimbers  is  equal  to  the  root  of 

the  dividend  divided  by  the  root  of  the  divisor, 

292.    Evolution  of  monomials. 

EXERCISES 

1.    Extract  the  square  root  of  36  a%^. 

Solution. — Since,  in  squaring  a  monomial,  §278,  the  coefficient  is 
squared  and  the  exponents  of  the  letters  are  multiplied  by  2,  to  extract 
the  square  root,  the  square  root  of  the  coefficient  must  be  found,  and 
to  it  must  be  annexed  the  letters  each  with  its  exponent  divided  by  2. 

The  square  root  of  36  is  6,  and  the  square  root  of  the  literal  factors 
is  a^b.    Therefore,  the  principal  square  root  of  36  a^b^  is  6  a^b. 

The  square  root  may  also  be  —  6  a^b,  since  —  6  a^b  x  —6a^b  =  36  a^b-. 


.'.  V36  a^b-^  =  ±  6  a^b. 

2.    Extract  the  cube  root  of  —  125  ,ify^. 

Solution,  \/—  12-5  x*^?/-^i  =  —  5  x-y',  the  real  root 


EVOLUTION  217 

To  find  the  root  of  an  integral  term  : 

RuLK.  — Extract  the  required  root  of  the  numerical  coefficient ^ 
unnex  to  it  the  letters  each  luith  its  exjmnent  divided  bij  the  index 
of  the  root  sought ^  and  prefix  the  proper  sign  to  the  resuU. 

Find  real  roots : 

3.  Va%V\  8.    ^  -  8  o«&»*.       13.    V(-m6»)«. 

4.  Va«6'V\  9.    V^^^2  x'y.    14.    \/(-  ai'by. 
6.    \/tt»Vy».             10.    Vi6^-.  15.    -Va}^!^. 

6.  <'i?^JA^.  11.    ^-a"6»aJ*.     16.    -^/'^^2Tpr^, 

7.  VxV^-^-  12.    \/-243»/'".      17.    -\/-128a"/i28 

8a?v« 


18.   Extract  the  cube  root  of 


27  m^n^ 


2 


^27  m'n"      >X27  m*n^        3mn*  3mn^ 

To  find  the  root  of  a  fractional  term : 

Rule.  —  Find  the  refjuired  root  of  both  numerator  and  denomi- 
nator and  prefix  the  proper  sign  to  the  resulting  fraction. 

Find  real  roots : 


20.    #E^.       23.    <256^.  26.     ^P^. 


\    243 «»        "•    V     17281-'  "•    \2"-^ 


ST" 


218 


EVOLUTION 


293.    To  extract  the  square  root  of  a  polynomial. 


1.    Find    the    process 
a2  +  2a6  +  &'. 


EXERCISES 

for    extracting    the    square    root   of 

PROCESS 


a'-\-2ab  +  b-\a-\-b 

r,2 


Trial  divisor,         2  a 
Complete  divisor,  2  a  +  b 


2  ab  +  // 
2ab  +  b' 


Explanation.  —  Since  a'^  -]-  2  ab  +  b'^  is  the  square  of  («  +  &),  we  know 
that  the  square  root  of  a'^  +  2  afo  -f  h'^  is  a  +  b. 

Since  tlie  first  term  of  the  root  is  a,  it  may  be  found  by  taking  the 
square  root  of  a^,  the  first  term  of  the  power.  On  subtracting  a'^,  there  is 
a  remainder  of  2  a&  +  h'^. 

The  second  term  of  the  root  is  known  to  be  ?>,  and  that  may  be  found 
by  dividing  the  first  term  of  the  remainder  by  twice  the  part  of  the  root 
already  found.     This  divisor  is  called  a  ti'ial  divisor. 

Since  2  ab  +  b^  is  equal  to  6(2a  +  &),  the  complete  divisor  v/hich 
multiplied  by  b  produces  the  remainder  2  ab  +  U^  is  2  a  +  6  ;  that  is,  the 
complete  divisor  is  found  by  adding  the  second  term  of  the  root  to  twice 
the  root  already  found. 

On  multiplying  the  complete  divisor  by  the  second  term  of  the  root 
and  subtracting,  there  is  no  remainder  ;  then,  «  +  &  is  the  required  root. 

2.    Extract  the  square  root  of  9  :x^  —  ^Oxy-\-  25  y\ 

PROCESS 


9  0^2 _  30  xy  +  25 y'\3x-5y 
9ar^ 


Trial  divisor 


6x 


Complete  divisor,  6  x  —  5  y 

Extract  the  square  root  of : 

3.  4:X^-{-12x-\-9. 

4.  x^-{-2x-\-l. 

5.  1  —  4  m  +  4  m^ 


-  30  xy  +  25  y^ 

-  30  xy  +  25  / 


6.  c2-12c-f-36. 

7.  4.^•2  +  4;c4-l. 

8.  16 +  24  .^'  +  9. 1-2. 


Since,  in  squaring  a  +  S  +  c,  a-\-b  may  be  represented  by  .i-, 
and  the  square  of  the  number  by  x^  +  2  xc  +  r,  the  square  root 


EVOLUTION  219 

of  a  number  whose  root  consists  of  inore  tJian  two  teiina  may  be 
xtracted  iu  the  same  way  as  in  exercise  1,  by  considering  the 
■>  nii»  already  found  as  one  term. 

9.    Find  the  square  root  of  4  a;*  -h  12  ar*  -  3  or^  - 18  a;  -f-  9. 

PROCESS 

4a^  +  12ar»-3g^-18a;  +  9|2a^  +  3a;-3 

4a^ 

4x* 


4ir*  +  3x 


12a:»-3a^ 
12ar»4-9a^ 


4a^  +  6x 
4a:*4-6a; 


-12ar^-18a;  +  9 
-12x2-18  a;-f-9 


Explanation.  —  Proceeding  as  in  exercise  2,  we  find  that  the  first  two 
terms  of  the  root  are  2x^  ■\-  'ix. 

Considering  (2  x'  +  3  x)  aa  the  firat  term  of  the  root,  we  find  the  next 
tenn  of  the  root  as  we  found  the  second  term,  by  dividing  the  remainder 
by  twice  the  part  of  the  root  already  found.  Hence,  the  trial  divisor  is 
t  J--2  +  6x,  and  the  next  term  of  the  root  is  -  3.  Annexing  this,  as  before, 
ti)  the  trial  divisor  already  found,  we  find  that  the  complete  divisor  is 
2  x2  4. 3  X  —  3.  Multiplying  this  by  -  3  and  subtracting  the  product  from 
—  12x*  —  18x  +  0,  we  have  no  remainder.  Hence,  the  square  root  of  the 
number  is  2  x-^  +  3  x  —  3. 

Rule.  —  Arrange  theterms  of  the  polynomial  with  reference  to 
(he  consecittire  poicers  of  some  letter. 

Extract  the  square  root  of  the  first  term,  icrite  the  result  as  the 
first  term  of  the  root,  and  aubtri"  f  ifs  square  from  the  given 
]>ol>i)tf>n}ial. 

hiri'l''  the  first  term  of  the  remainder  by  tirice  the  root  already 
found,  used  cw  a  trial  dioisor,  and  the  quotient  will  be  the  next 
term  of  the  root.  Write  this  result  in  the  root,  and  annex  it  to 
the  trial  divisor  to  form  the  complete  divisor. 

Multiply  the  complete  divisor  by  this  term  of  the  rooty  and.  stib- 
tract  the  product  from  the  first  remainder. 

Find  the  next  term  of  the  root  by  dividing  the  first  term  of  tlie 
remainder  by  the  firat  term  of  the  trial  divisor. 

Form  the  complete  divisor  as  before  and  continue  in  this  man- 
ner until  all  the  terms  of  the  root  have  been  found. 


220  EVOLUTION 

Extract  the  square  root  of : 

10.  25«2_40a-M6.  13.  4  a;^  -  52  a;-^ -f  169. 

11.  900ar^^-60.^•^-l.  14.  ^d^-^cP,i' -\- y^n\ 

12.  x^ -\- xy -\- {  y^.  15.  (a  + 6)- -  4  (a  + /^) +4. 

16.  9x'-12x^-i-10x'-Ax-i-l. 

17.  x'-6x^y-\-13xY--12xy"-i-4:y\ 

18.  x^  +  2  a V  -  a V  -  2  a V  +  al 

19.  25a;^H-4-12a;-30ar^  +  29a;l 

20.  l-2x+3aj2_4^3_^3^4_2ar^^a;«. 

21.  a^-2a-6  +  2aV-26c24-62^_,.4. 

22.  4  a^  -  12  a6  +  16  ac  +  9  6^  +  16  c^  -  24  be. 

23 .  9  a;-  -h  25  ^2  +  9  ^2  _  30  a;y  +  18  xz  -  30 1/^. 

24.  ^  +  ^'+25. 
9         3 

25.  ^+15  + 9  711 

4  71- 

26.  ^-^  +  4. 
16  7-'       r 

30 .    i^s  +  4  .T^  -  3  a-^  -  20  .a;^  -  2 .«« +  4  -h  4  a?"  - 1 6  a,'  -f  32  a^. 


27. 

u^i-^'- 

28. 

a;^  +  2a^-l---h^. 
X      X- 

29. 

■■'-''-If-i-k 

31. 


4a;^      4ar^      Sx^      2x 

y*      f       y^      y 


__     a^  .     o     ,  4a^a;-  ,  2aa;'^  ,  x^ 

32.  _.  +  a3.  +  -^  +  -^+-. 

33.  l-^-Vf +  4a^-2«6  +  f. 

4m^ _  i^^  .  1  ^ ^^i'  .  §J^  _  '^'l^^i'  ■  3m       9 
■       9  3  15  5  50         10      16  * 

35.      ,^  _  4  ^.7  _|_  _4^  ^,6  +  I  ^^  _  2_0  ^4  _^   1^2  ^.3  _^  2  5  ./2  _  5  ,,  _^  9^ 


EVOLUTION  221 

36.   Find  four  terms  of  the  square  root  of  1  -|-  x. 
Solution 

1 


2-\-kx 


2  +  X-i[3i^ 


X 


-   \x^-\^  +  ifjX* 


Find  the  square  root  of  the  following  to  four  terms : 

37.  1-a.  39.    .r-1.  41.    .V'-fS. 

38.  a- 4-1-  40.   4-(t.  42.    a^-}-2b. 

SQUARE    ROOT    OF    ARITHMETICAL   NUMBERS 

294.  Compare  the  number  of  digits  in  the  square  root  of 
ciich  of  the  following  numbers  with  the  number  of  digits  in  the 
mimher  itself: 


M  MHKR 

KOOT 

NKMHKK 

ROOT 

NUMKKK 

ROOl 

1 

1 

roo 

10 

I'OO'OO 

100 

25 

5 

10'24 

32 

56'2o'()0 

750 

81 

9 

98'01 

99 

99'80'01 

999 

From  the  preceding  comparison  it  may  be  observed  that: 

Vkincii'LK.  —  If  a  number  is  separated  into  periods  of  two 
'liifits  each,  beginning  at  nnits,  its  square  root  will  have  as  many 
(h'fjits  as  the  number  ha^  periods. 

The  left-hand  period  may  be  incomplete,  consisting  of  only  one  digit. 

295.  If  the  number  of  units  expressed  by  the  tens'  digit  is 
represented  by  t  and  the  number  of  units  expressed  by  the 
units'  digit  by  ?<,  any  number  consisting  of  tens  and  units  will  be 
represented  by  1 4-  u,  and  its  sfjuare  by  (t  +  uf,  or  t^-\-2tu-\-  u\ 

Since  25  =  20  +  6,  252  =  (20  +  5)^  =  20^  +  2  (20  x  6)  +  5*  =  625. 


222  EVOLUTION 

EXERCISES 

296.    1.   Extract  the  square  root  of  3844. 

FIRST    PROCESS 


f  = 


2  i  =  120 
u=     2 


2t  +  u=122 


38'44l60  +  2  Explanation. — Separating    the 

o/?  AA  number  into  periods  of  two   digits 

each  (Prin.,  §  294),  we  find  that 
the  root  is  composed  of  two  digits, 
tens  and  units.     Since  the   largest 


2  44 

2  44  square  in  38  is  6,  the  tens  of  the  root 


cannot  be  greater  than  6  tens,  or  60. 
Writing  6  tens  in  the  root,  s(iuaring,  and  subtracting  from  3844,  we 
have  a  remainder  of  244. 

Since  the  square  of  a  number  composed  of  tens  and  units  is  equal  to 
(the  square  of  the  tens)  +  {twice  the  product  of  the  tens  and  the  units)  + 
(the  square  of  the  units),  when  the  square  of  the  tens  has  been  subtracted, 
the  remainder,  244,  is  twice  the  product  of  the  tens  and  the  units,  plus 
the  square  of  the  units,  or  only  a  little  more  than  twice  the  product  of  the 
tens  and  the  units. 

Therefore,  244  divided  by  twice  the  tens  is  approximately  equal  to  the 
units.  2x6  tens,  or  120,  then,  is  a  trial,  or  partial,  divisor.  On  dividing 
244  by  the  trial  divisor,  the  units'  figure  is  found  to  be  2. 

Since  twice  the  tens  are  to  be  multiplied  by  the  units,  and  the  units 
also  are  to  be  multiplied  by  the  units  to  obtain  the  square  of  the  units,  in 
order  to  abridge  the  process  the  tens  and  units  are  first  added,  forming 
the  complete  divisor  122,  and  then  multiplied  by  the  units.  Thus, 
(120  +  2)  multiplied  by  2  =  244. 

Therefore,  the  square  root  of  3844  is  62. 

SECOND    PROCESS 

38'44'62  Explanation. — In    practice    it 

•2 '  o^  is  usual  to  i)lace  the  figures  of  the 

same  order  in  the  same  column,  and 
to  disregard  the  ciphers  on  the  right 
of  the  products. 


2t  =  V2() 


2  t-\-u  =  122 


2  44 
2  44 


Since  any  number  may  be  regarded  as  composed  of  tens  and 
units,  the  foregoing  processes  have  a  general  application. 

Thus,  346  =  34  tens  +  6  units  ;  2377  =  237  tens  +  7  units. 


EVOLUTION  223 

2.    Extract  the  square  root  of  104976. 


Solution 


Trial  divisor  =  2  x    30  =  60 

Complete  divisor  =  60  +    2  =  62 


i(»'40'7(;[;{24 

9 

149 
124 


Trial  divisor  =  2  x  320  =  (540 

Complete  divisor  =  640  -f  4  =  044 


25  70 
25  76 


Rule.  —  Separate  the  nvmher  into  periods  oftirojigures  eachy 
I  "(J  inning  at  units. 

Find  the  greatest  square  in  the  left-hand  period  and  write  its 
root  for  the  first  figure  of  the  required  root. 

Square  this  i-oot,  subtract  the  result  from  the  left-hand  period^ 
and  annex  to  the  remainder  the  next  jjcn'ofJ  0,,-  a  new  dividend. 

Double  the  root  already  found,  with  a  cipher  annexed,  for  a 
trial  divisoi;  and  by  it  divide  the  dividend.  Tfie  quotient,  or 
quotient  diminished,  ivill  be  the  second  figure  of  the  root.  Add 
to  the  trial  divisor  the  figure  last  found,  multiply  this  complete 
divisor  by  the  figure  of  the  root  laM  found,  subtract  the  product 
from  the  dividend,  and  to  the  remainder  annex  .the  next  period 
for  the  next  dividend. 

Proceed  in  this  m^mner  until  all  the  periods  have  been  used. 
The  res II J f  irilj  be  the  square  root  sought. 

1.  Wlieii  tlie  number  is  not  a  perfect  square,  annex  periods  of  decimal 
ciphers  and  continue  the  process. 

2.  Decimals  are  pointed  off  from  the  decimal  point  toward  the  right. 

3.  The  square  root  of  a  common  fraction  may  be  found  by  extracting 
the  stiuare  root  of  both  luimerator  and  denominator  separately  or  by 
reducing  the  fraction  to  a  decimal  and  then  extracting  the  root. 

Extract  the  square  root  of : 

3.  o29.                          6.  57121.  9.  2480.04. 

4.  2209.                        7.  42025.  10.  10.9561. 

5.  4761.                        8.  95481.  11.  .001225. 


54 

EVOLUTION 

12. 

186624. 

13.  1332.25. 

14. 

111.0916. 

15. 

m- 

17. 

ill              19-  Ml- 

21.  m. 

16. 

m- 

18. 

TV¥(r-              20.  Ifl 

22.  Iff. 

Extract  the  square  root  to  four  decimal  places : 

23.  |.  25.  |.  27.  |.  29.  f 

24.  |.  26.   .6.  28.  |.  30.   ^^ 
297 .    To  extract  the  cube  root  of  a  polynomial. 

EXERCISES 

1.    Find  the  process  for  extracting  the  cube  root  of  a^-\-Sa^b 
+  3ab'-i-  b^ 

PROCESS 

a^ ^3  a'b  -\-8  ab''  +  b'\a_±b 

a? 
Trial  divisor,  3  a? 

Complete  divisor,  3  a-  +  3  ab  +  b'^ 


3d'b-{-3ab''-\-l/' 
3  a^b  +  3  o6^  +  b^ 


Explanation.  — Since  d^  +  3a-&  +  3  ah"^  +  h^  is  the  cube  of  (a  +  6), 
we  know  that  the  cube  root  of  a^  +  3  d^h  +  3  ah'^  +  6Ms  «  +  &. 

Since  the  first  term  of  the  root  is  a,  it  may  be  found  by  taking  the 
cube  root  of  a^,  the  first  term  of  the  power.  On  subtracting,  there  is  a  re- 
mainder of  3  a%  +  3  ah'^  +  b^- 

The  second  term  of  the  root  is  known  to  be  6,  and  that  may  be  found 
by  dividing  the  first  term  of  the  remainder  by  3  times  the  square  of  the 
part  of  the  root  already  found.     This  divisor  is  called  a  tnal  divisor. 

Since  3  d^h  +  3  ah^  +  6"^  is  equal  to  h  (3  d^  +  3  c?ft  +  6^),  the  complete 
divisor,  which  multiplied  by  h  produces  the  remainder  3  a^ft  +3  ah'^  +  &^ 
is  3  a^-fS  aft  -f  ft2  .  that  is,  the  complete  divisor  is  found  by  adding  to  the 
trial  divisor  3  times  the  product  of  the  first  and  second  terms  of  the  root 
and  the  square  of  the  second  term  of  the  root. 

On  multiplying  the  complete  divisor  by  the  second  term  of  the  root,  and 
on  subtracting,  there  is  no  remainder  ;  then,  a  +  6  is  the  required  root. 

Since,  in  cubing  a-\-b  +  c,  a-\-b  may  be  expressed  by  Xy  the 
cube  of  the  number  will  be  x^ -^3  x-c  +  3xc^-\-(^.  Hence,  it  is 
obvious  that  the  cube  root  of  an  expression  whose  root  consists 
of  more  than  two  terms  may  be  extracted  in  the  same  way  as  in 
exercise  1,  by  considering  the  terms  already  found  as  one  term. 


EVOLUTION  226 

2.   Fiiid  the  cube  root  of  6«  -3  ft*  +  56*-  3  6  - 1. 


PROCESS 


6«_3  b'+  5  //'^-3  b-l\b''-b-l 
b' 


Trial  divisor,  3b* 

( 'oiuplete  divisor,     36^—36*^+6' 


-36^  +  56^ 
-36^-f36^-6» 


I'rial  divisor,  3 6^ -G 6'' +36=* 

Complete  divisor,    36^— G6'' +36  +  1 


-36^  +  66''-36-l 
-36*  +  66«-36-l 


Explanation.  —  The  first  two  terms  are  found  in  the  same  manner 
as  in  the  previous  exercise.  In  finding  the  next  term,  b-  —  b  ia  con- 
sidered as  one  term,  which  we  square  and  nniltiply  by  3  for  a  trial 
divisor.  On  dividing  the  remainder  by  this  trfal  divisor,  the  next  term 
of  the  root  is  found  to  be  —  1.  Adding  to  the  trial  divisor  3  times 
(^2  _  ft)  multiplied  by  —  1,  and  the  square  of  —  1,  we  obtain  the  com- 
plete divisor.  On  multiplying  this  by  —  1,  and  on  subtracting  the  product 
from  —  3  6*  +  6  6''  —  3  6  —  1,  there  is  no  remainder.  Hence,  the  cube 
root  of  the  polynomial  is  6^  —  6—1. 

Rule.  —  An'ange  the  polynomial  with  reference  to  the  consecu- 
tive powers  of  some  letter. 

Extract  tJie  cube  root  of  the  first  terniy  write  the  result  as  the 
first  term  of  the  rooty  and  subtract  its  cube  from  the  given 
polynomial. 

Divide  the  first  term  of  the  remainder  by  three  times  the  sqvxire 
of  the  root  already  founds  used  as  a  trial  divisor,  and  the  quotient 
imll  be  the  next  term  of  the  root. 

Add  to  this  trial  divisor  three  times  the  product  of  the  first  and 
sirnn'l  iri'ms  of  the  root,  and  tlie  square  of  the  second  term.  The 
ri'nult  will  be  the  complete  divisor. 

Multiply  the  complete  divisor  by  the  last  term  of  the  rootfownd, 
and  subtract  this  product  from  the  dividend. 

Find  the  next  term  of  the  root  by  dividing  the  first  term  of  the 
remainder  by  the  first  term  of  the  trial  divisor. 

Form  the  compiete  dixnsor  as  before,  considering  the  part  of  the 
root  already  found  as  the  first  term,  and  continue  in  this  manner 
until  all  the  terms  of  the  root  are  found. 
milnk's  stand,  alg.  — 16 


226  EVOLUTION 

Find  the  cube  root  of : 

3.  x^  —  3  x-y  +  3  xy-  —  }f. 

4.  m"^  -  9  7)124. 27  m- 27. 

5.  8  m'  -  60  m'li  +  loO  mir  -  125  n^. 

6.  27ar''-189a;->  +  441a'/-343/. 

7.  125  a^  +  675  a2a;  + 1215  aa;2  + 729  0^3. 

8.  1000  p^  -  300  p^g  +  30  p-q-  -  r/. 

9.  m^  +  6  m^  4- 15  m*  +  20  ?/i^  +  15  wi-  +  6  m  +  1. 

10.  a:«-6ar^  +  i5.r4-20ar^  +  15a^-6a;  +  l- 

11.  a;*^4-3a;5  +  9a;^  +  13ar^  +  18a;2  4.-^2x  +  8. 

12.  a;*'  +  12  t'  +  (j3x'-\-  184  ar^  +  315  a^  +  300  a;  + 125. 

13.  a;«  4-  6  a^  -  18  ic*  -  1000  +  180  a.-^  -  112  ar^  4-  600  x. 

14.  l-6a  +  21a2-44a3  +  63a^-54a^  +  27a«. 

15.  8  n^  +  42  ?i^  -  9  n«  4-  36  n*  H-  9  n«  -  21  n^  -  n\ 

16.  a^3_i2ar^  +  54a^-112  +  — -^  +  ^. 

,  „     o-^6^a;^     cV  ,  3  acx^     3  a-/>.T*^ 
&  ¥  0  c 

18.    aj«  +  15ar^4-i?  +  20  +  ^  +  -  +  6x*. 
a;^  x^     x*' 

ar*     2ar'     4a,-      8         2 

8  n-  n  2        7r^ 

21.    c«  -  3  c^fZ  -  3  c''^?^  _j_  1 1  ,.v^3  _^  (5  ^.2^^4  _  -^2  cf?^  -  8  (E 


EVOLUTION 


227 


22.    L^-^4-5^-45  +  ??-*-^  +  ^. 

if^  iT  8  r  7^  7^ 

2i       oar     or     ar 

J  4  o  J  o  4  Z 

25.  ^-V'^  +  ^'+^V^AV'l^V?^. 
a«      a'  ^  a'  ^  a'  ^  a'«  ^  a"      a'» 

26.  7i«  -  f  n«  +  i  n*-  l^?i»H-  f  7i»-  f  rj  +  J. 

27.  3ji^r«_  Jr*_j/*  + 2^3^3-1- 5jir2_^-r- 27. 


CUBE   ROOT  OF  ARITHMETICAL  NUMBERS 

298.    Compare  the  number  of  digits  in  each  number  and  in 
its  cube  root : 


UMBER 

ROOT 

NUMBER 

ROOT 

NUMBER 

ROOT 

1 

1 

I'OOO 

10 

I'OOO'OOO 

100 

27 

3 

27'000 

30 

27'000'000 

300 

729 

9 

970'299 

99 

997'002'999 

999 

Observe  that : 

Principle.  —  If  a  number  is  separated  into  periods  of  three 
digits  eachf  beginning  at  units,  its  cube  root  will  have  as  many 
digits  Gw  the  number  has  periods. 

The  left-hand  period  may  be  incomplete,  consisting  of  only  one  or  two 

digits. 

299.  If  the  number  of  units  expressed  by  the  tens'  digit  is 
represented  by  <,  and  the  number  of  units  expressed  by  the 
units*  digit  is  represented  by  m,  any  number  consisting  of  tens 
and  units  will  be  represented  by  <  -f  «,  and  its  cube  by  (t  +  u)', 
or  f^  +  S  t^u  4-  3  tu^  +  u\ 

Thus,  25  =  2  t^ns  +  6  luiits,  or  (20  +  5)  units, 

and  258  =  20»  +  3(20'^  x  6)  +  3(20  x  S*)  +  5«  =  16625. 


228  EVOLUTION 

EXERCISES 

300.    1.   Extract  the  cube  root  of  12167. 


FIRST    PROCESS 

12'167  [20  +  3 
f=  8  000 


3^^  =  1200 

Ztu^z    180 

u^=       9 

4167 

=  1389 

4167 

Trial  divisor, 


Complete  divisor, 

ExPLAKATiON.  —  On  Separating  121(57  into  periods  of  three  figures  each 
(§  298,  Prin.)  there  are  found  to  be  two  digits  in  the  root,  that  is,  the 
root  is  composed  of  tens  and  units.  Since  the  cube  of  tens  is  thousands, 
and  the  thousands  of  the  power  are  less  than  27,  or  S^,  and  more  than  8, 
or  2^,  the  tens'  figure  of  the  root  is  2.  2  tens,  or  20,  cubed  is  8000,  and  8000 
subtracted  from  12167  leaves  4167,  which  is  equal  to  3  times  the  tens^  x 
the  units  +  3  times  the  tens  x  the  units"-^  +  the  units^. 

Since  3  times  the  tens^  x  the  units  is  much  greater  than  3  times  the 
tens  X  the  units'-^  +  the  units-^,  4167  is  only  a  little  more  than  3  times  the 
tens^  X  the  units.  If,  then,  4167  is  divided  by  3  times  the  tens^,  or  by 
1200,  the  trial  divisor,  the  quotient  will  be  approximately  equal  to  the 
units,  that  is,  3  will  be  the  units  of  the  root,  .provided  proper  allowance 
has  been  made  for  the  additions  necessary  to  obtain  the  complete  divisor. 

Since  the  complete  divisor  is  found  by  adding  to  3  times  the  tens'^ 
the  sum  of  3  times  the  tens  x  the  units  and  the  units^,  the  complete 
divisor  is  1200  +  180  +  9,  or  1389.  This  multiplied  by  3,  the  units,  gives 
4167,  which,  subtracted  from  4167,  leaves  no  remainder. 

Therefore,  the  cube  root  of  12167  is  20  +  3,  or  23. 

SECOND    PROCESS 

12'167  I  23  Explanation.  —  In  practice  it  is  usual  to 

f   =  8  place  figures  of  the  same  order  in  the  same 

column,  and  to  disregard  the  ciphers  on  the 
right  of  the  products. 


3^2  =1200 

3tu=    180 

u^=       9 

4167 

1389 

4167 

Since  a  root  expressed  by  any  num- 
ber of  figures  may   be   regarded    as 
composed  of  tens  and  units,  the  pro- 
cesses of  exercise  1  have  a  general  application. 
Thus,  120  -  12  tens  +  0  units ;  1203  =  120  tens  +  3  units. 


EVOLUTION  229 


2.   Extract  the  cube  root  of  1740992427. 
Solution 


1'740'992'427   |  1203 

1 


rst^  =3(10)2 

|3<u  =3(10x2) 

=      300 

60 

4 

740 

304 

728 

Sfi  =3(120)2 

=      43200 

12992 

Hfi  =3(1200)2 
Stu  =  3(1200  X  3) 
u2  =  32 

=  4320000 

=      10800 

9 

12J>U2427 

4330801 

) 

12992427 

Since  the  third  figure  of  the  root  is  0,  it  is  not  necessary  to  form  the 
complete  divisor,  inasmuch  as  the  product  to  be  subtracted  will  be  0. 

Rule.  —  Separate  the  number  into  j)eriods  of  three  figures  each, 
bey  inning  at  units.  Find  the  greatest  cube  in  the  left-hand  per  iody 
and  write  its  root  for  the  first  digit  of  the  required  root. 

Cube  this  root,  subtract  the  result  from  the  left-hand  period, 
and  annex  to  the  remainder  the  next  period  for  a  new  dividend. 

Take  three  times  the  square  of  the  root  already  found,  annex 
two  ciphers  for  a  trial  divisor,  and  by  the  result  divide  tJie  divi- 
dend. Tfie  quotient^  or  quotient  diminished,  tvill  be  the  second 
figure  of  the  root. 

To  this  trial  divisor  add  three  times  the  jiroduct  of  the  first  part 
of  the  root  with  a  cipher  annexed,  multiplied  by  the  second  part, 
and  also  the  square  of  the  second  part.  Tlieir  sum  will  be  the 
complete  divisor. 

Multiply  the  complete  divisor  by  the  second  part  of  the  root,  and 
subtract  the  product  from  the  dividend. 

Continue  thus  until  all  the  figures  of  the  root  have  been  found. 

1.  When  there  is  a  remainder  after  subtracting  the  last  product,  annex 
decimal  ciphers,  and  continue  the  process. 

2.  Decimals  are  pointed  off  from  the  decimal  point  toward  the  right. 

3.  The  cube  root  of  a  common  fraction  may  be  found  by  extracting  the 
cube  root  of  the  numerator  and  the  denominator  separately  or  by  reduc- 
ing the  fraction  to  a  decimal  and  then  extracting  its  root. 


JO 

EVOLUTION 

Extract  the  cube  root  of : 

3.  29791. 

9.  2406104. 

15. 

.000024389. 

4.  54872. 

10.  69426531. 

16. 

.001906624. 

5.  110592. 

11.  28372625. 

17. 

.000912673. 

6.  300763. 

12.  48.228544. 

18. 

.259694072. 

7.  681472. 

13.  17173.512. 

19. 

926.859375. 

8.  941192. 

14.  95.443993. 

20. 

514500.058197. 

Extract  the  cube  root  to  three  decimal 

places 

21.  2. 

23.  .8. 

25. 

5 
04' 

27.  i. 

22.  5. 

24.  .16. 

26. 

|. 

28.  ^V 

ROOTS  BY  VARIOUS   METHODS 
301.   By  inspection  and  trial. 

To  find  the  cube  root  of  a  number,  as  343,  we  estimate  the 
root  and  cube  it.  If  the  cube  is  greater  or  less  than  the  num- 
ber, our  estimate  must  be  modified,  for  the  cube  of  the  root 
must  be  the  number  itself. 

This  method,  which  is  the  general  one  in  evolution,  may  be 
used  to  find  any  root  of  a  polynomial. 


By  inspection  we  estimate  \/x^  —  lOx*  +  40x3  —  80  a:-  +  80  a:  —  82  to 
be  X  —  2,  'noting  the  number  of  terms  and  the  first  and  last  terms. 
By  trial  x  —  2  proves  to  be  the  root,  for  its  fifth  power  is  found  to  be 
the  given  polynomial. 

302.   By  factoring. 

This  method  consists  in  factoring,  grouping  the  factors,  and 
extracting  the  root  of  each  group. 

Thus,        v/42875  =  ^5  •  5  •  5  •  7  •  7  •  7  =  \/53  .  7^  =  5  •  7  =  35; 
also,      Vx4  +  2a;3_3x2_4x  +  4  =  V(x  -  l)2(x  +  2)2  =  (x  -  1 ) (x  +  2) 
=  x2  -f-  X  -  2. 


KVOLL HON 


'S61 


303.   By  successive  extraction  of  roots. 

Since  the  fourth  power  is  the  square  of  the  second  power,  the 
sixth  power  the  cube  of  the  second  power,  etc.,  any  indicated 
r  H't  w  ii.KO  index  is  4,  6,  8,  '.).  •  !(  .,  may  be  found  by  extracting 
successively  the  roots  corres[)()n(ling  to  tlie  factors  of  the  index. 

The  fourth  root  may  be  obtained  by  extracting  the  square  root  of  the 
luare  root ;  the  sixth  root,  by  extracting  the  cube  root  of  the  square  root, 
1  the  square  root  of  the  cube  root;  the  eighth  root,  by  extracting  the 
iuare  root  of  the  square  root  of  the  square  root. 


EXERCISES 

304.  Using  any  method,  find  the : 

1.  Square  root  of  «« -  12  a^  -f-  36. 

2.  Cube  root  of  125  —  7.")  x  -\-  1 .")  .r-  —  x\ 

3.  Fourth  root  of  If)  -  32  j;  -|-  24  x-  -  8  .r^  -f  x\ 

4.  Fourth  root  of  x'  +  12  a:»y  +  54  x^y-  +  108  xif  +  81  y\ 

5.  Fourth  root  of  16  m^  -  32  iv?  +  24  m^  -  8  m  +  1. 

6.  Fifth  root  of  32ar'  +  80a^  +  80x»  +  40x2  +  10a;  +  l. 

7.  Fifth  root  of  a»«-f  15  a*  +  90  a«  -h  270  a*  +  405  a?-\-  243. 

Find  the  sixth  root  of : 

8.  ./-12x'-f  60a:^-160a^  +  240a:*-192a;H-64. 

9.  64ic«-576u:«-f  2160  0:^-4^^20  ar»  + 4860  a:2_  2916  a; +  729. 
10.   a:*  -h  6  OCX*  + 15  a?(?Qi^  +  20  o'c^j?  + 15  « W  +  6  aV  x  +  aV. 


Find  the  indicated  root : 

11.    <^^375.  15.    ^262144. 


19.    </4084101. 


12.  -1/1296. 

13.  -i' 50625. 

14.  ^4605f). 


16.  V759;i75. 

17.  v/531441. 

18.  v^576480l. 


20.  V16777216. 

21.  V^137569. 

22.  ^10604499373. 


THEORY   OF   EXPONENTS 


305.  Thus  far  the  exponents  used  have  been  positive  integers 
only,  and  consequently  the  laws  of  exponents  have  been  obtained 
in  the  following  restricted  forms : 

1.  a""  xa''^  a'"^"  when  m  and  n  are  positive  integers. 

2.  a"*  H-  a"  =  a"*-"  when  m  and  n  are  positive  integers  and 
m  is  greater  than  ii. 

3.  («"*)"=!«'""  when  m  and  n  are  positive  integers. 

4.  Va"*  =  ci"*^"  when  m  and  7i  are  positive  integers,  and  m  is 
a  multiple  of  n. 

5.  (cibY  —  ct^'ft**  when  n  is  a  positive  integer. 

If  all  restrictions  are  removed  from  m  and  n,  we  may  then 
have  expressions  like  a~^  and  a^ .  But  such  expressions  are 
as  yet  unintelligible,  because  —  2  and  -|  cannot  show  how 
many  times  a  number  is  used  as  a  factor. 

Since,  however,  these  forms  may  occur  in  algebraic  processes, 
it  is  important  to  discover  meanings  for  them  that  will  allow 
their  use  in  accordance  with  the  laws  already  established,  for 
otherwise  great  complexity  and  confusion  would  arise  in  the 
processes  involving  them. 

Assuming  that  the  law  of  exponents  for  multiplication, 

a*"  X  a"  =  a'"+", 
is  true  for  all  values  of  m  and  n,  the  meanings  of  zero,  negative, 
and  fractional  exponents  may  be  readily  discovered  by  substi- 
tuting these  different  kinds  of  exponents  for  m  and  n  or  both, 
and  observing  to  what  conclusions  we  are  led. 

2.S2 


THEORY  OF   EXPONENTS  233 

306.  Meaning  of  a  zero  exponent. 

We  have  agreed  that  any  new  kind  of  exponent  shall  have 
its  meaning  determined  in  harmony  with  the  law  of  exponents 
for  multiplication,  expressed  by  the  formula, 

a'"  X  a"  =  a"*"*"*. 

If  n  =  0,  a""  xaP=  a'""^**,  or  a"*. 

Dividing  by  a*".  Ax.  4,  a^'  =  ^  =  1.     That  is. 

Any  7iumber  with  a  zero  exponent  is  equal  to  1. 

307.  Meaning  of  a  negative  exponent. 

Since,  §  305,  a"*  x  a"  =  a"'^",  is  to  hold  true  for  all  values  of 
m  and  ?i,  if  m  =  —  n. 


if  m  =  —  n. 

a""  X  a''  =  a~*+" 


But,  §306,  a«  =  l. 

Hence,  Ax.  5,  a""  x  a"  =  1. 

Dividing  by  a",  Ax.  4,         a~''=  —  •     That  is, 

a" 

Any  number  with  a  negative  exponent  in  equal  to  the  reciprocal 
of  the  same  number  with  a  numerically  equal  positive  exponents 

308.    Hy  the  definition  of  negative  exponent  just  given, 

a-'-  =  —  and  6-"  =  ^-- 
a"*  6* 

1^ 

Therefore,  ^  =  ^  =  i-  X  ^'  =  — .     Hence, 

6—      J^      a*      1      a* 

PRINCIPLK.  —  Any  factor  may  he  transferred  from  one  term 
of  a  fraction  to  the  other  without  changing  the  value  of  the  frac- 
tiony  provided  the  sign  of  the  exponent  is  changed. 


234  THEORY    OF   EXPONENTS 

EXERCISES 

309.    Find  a  simple  value  for : 

1.  5«.                  3.    2-K               5.    (-3)'l  7.    {a^h'^qf, 

2.  4-1               4.   3-1               6.    (-6)-2.  '    8.    (-i)-^ 
9.    Which  is  the  greater,  {Vf  or  {Vfl  (i)  ^  or  (i)'^? 

10.  Find  the  value  of  2^  -  3  •  2-  +  5  •  2>  -  7  •  2«  +  4  •  2^1  -  2-2. 

11.  Find  the  value  of  a^  — 3a;^  +  4  ic^  +  a;"^  —  oaj-^  +  a;-^  when 
a;=L|  whena;  =  — |;  when  a;  =  1. 

12.  Which  is  the  greater,  (-|)-3or  (i)^?  (-|)-''or  (i^? 
Write  with  negative  exponents : 

13.  1h-5.  15.    1h-2\  17.    c^a^x^. 

14.  \-=rcC\  16.    a-r-ic^.  18.    am^-J-fex". 

19.  Write  5  ic~y  with  positive  exponents. 

Solution.  —  By  §  307,       5  x-^y"^  =  5y^—  =  ^. 

x-^      x^ 

Write  with  positive  exponents : 

20.  2x-\  23.    a-''b-\  26.  4  aV^. 

21.  5a-^  24.    x-^y-\  27.  3  aa;-l 

22.  3  6-2.  25.    a-^6V^  28.  a"6-^". 

29.  4.x^-2x^-{-5x'-6x^  +  Sx-^-5x-\ 

30.  2a3-12a2-16a  +  3a«-f 2a-i-7a-l 

31.  a'b-''  -  a'b-'-  +  aft-^  -  1  +  cr'b  -  a-'-b~  +  a-^6^ 

32.  Write  ^  without  a  denominator. 

b:r 

Solution.  —  By  §  808,  '1^  =  3  a^fe-ix'^y. 

Write  without  a  denominator : 

33.  ^•.  34.     '^.  35.     ^,.  36.     f^Y- 


THEORY   OF    EXPONENTS  235 


37.     ^^.         40.     -i^.  43.     4^.  46. 


©•■ 


38.     «^'"t'.       41.    ^^^'-         44.    5:!.  47.     ^. 


39.     ±^.  42.     2i?!5j.        45.    f^y.  ,8. 


(a6)« 


310.    Meaning  of  a  fractional  exponent. 

Since  (§  305)  the  first  law  of  exponents  is  to  hold  true  for 

all  exponents,  .        .        .   , 

a^  Xa^  =  a*^*  =  a^  =  a; 

that  is,  a^  is  one  of  the  two  equal  factors  of  a,  or  is  a  square 
root  of  a.     The  other  square  root  of  a  is  —  a^ 

Again,  a^  xa^  =  a^^^  =  a* ; 

that  is,  a^  is  a  square  root  of  the  cube  of  a. 
Or,  similarly,    a^  xa^  Xa^  =  a^'*'^'^^  =  a^ ; 

that  is,  a^  is  the  cube  of  a  square  root  of  a. 

In  general,  confining  the  discussion  to  principal  roots,  let 
p  and  q  be  any  two  positive  integers,     l^y  the  first  law  of 


p^p. 


o  9  terniH 

exponents,  §  305,     a» •  f/«  •  •  •  to  9  factors  =  a«    "  --=  a". 

Therefore  a«,  one  of  the  q  equal  factors  of  a'',  is  a  qt\\  root 
of  the  pth.  power  of  a. 

Similarly,  a*  is  a  7th  root  of  a. 

*1  l  +  1+...top termii  ?        ''  . 

Also,  since  o« -a'   ••  to/)  factors  =  ««   *  =a',  a«is 

the  pth  power  of  a  qth  root  of  a. 

The  numerator  of  a  fractional  exponent  with  positive  integral 
terms  indicates  a  power  arid  the  denominator  a  root. 

The  fraction  as  a  whole  indicates  a  root  of  a  power  or  a  power 
of  a  root. 


236  THEORY  OF   EXPONENTS 

311.    Any  fractional  exponent  that  does  not  itself  involve  a 
root  sign  may  be  reduced  to  one  of  the  forms  ^  or  _-^. 

Thus,  8   2=8    ' . 


By  §§  307,  291, 


EXERCISES 

312.    1.    Find  the  value  of  16*. 

FiKST  Solution.     16?  =  VW  =  \/lG  .  16  •  16 

=  </(2.2  .2.2)(2.2  -2.  2)  (2.  2.  2  .2) 
=  \/(2  .2  .2)  (2  .2  .2)  (2  .2  .2)(2.2.2) 
=  2.2-2  =  8. 

Second  Solution.      16^  =  (16^)3  =  2^  =  8. 

In  numerical  exercises  it  is  usually  best  to  extract  the  root  first. 

Simplify,  taking  only  principal  roots : 

2.  8i 

3.  81 

4.  8-i 

5.  (-8)i 

14.  Which  is  the  greater,  27^  or  (-27)"^? 

15.  Which  is  the  greater,  (i)"  or  (\)~^  ? 

16.  Which  is  the  greater,  64-3  or  (Jj)!  ?  64t  or  (^L)"!  ? 

17.  Find  the  value  of  x" ^  _ 4  a?" ^  +  4  when  x=  —  y^. 


6. 

64l 

10. 

64-t. 

7. 

32l 

11. 

(syi 

8. 

2ot. 

12. 

(-32)-^. 

9. 

8II. 

13. 

16-1 

18.   Express  -Va-bc*  with  positive  fractional  exponents. 

2  1 
Solution.  Va'^bc-"^  =  aH^c'^  =  ^^^• 


THEORY  OF  EXPONENTS  237 

Express  with  positive  fractional  exponents : 

19.  Val?.  22.    i^/xf.  25.  (\/xy)-\ 

20.  Vxy.  23.    (y/yy.  26.  SVa;"^"*. 

21.  V*^.  24.    (</^f.  27.  2v^(a  +  6)^ 

Express  roots  with  radical  signs  and  powers  with  positive 
exponents: 

28.  ai  30.   xi  32.    x^yK  34.   a^-hx^, 

29.  x^.  31.    a^6i  33.    ah~^.  35.    x^-i-2/^. 

36.  Simplify  ^^  +  a?*  +  8^  +  3  a;^  -  d-s/x  -  y/2V, 

37.  Simplify  4^/x4-6  .i^-  3  a;"*  +  2^/x^-  8?  -  2  xi 

313.  Operations  involving  positive,  negative,  zero,  and  fractional 
exponents. 

Since  zero,  negative,  and  fractional  exponents  have  been 
defined  in  conformity  with  the  law  of  exponents  for  multiplica- 
tion, this  law  holds  true  for  all  exponents  so  far  encountered. 
For  the  proofs  of  the  generality  of  the  other  laws  of  exponents, 
see  the  author's  Academic  Algebra. 

EXERCISES 

314.  Multiply: 

1.  a»  by  a-*.  3.   a*  by  a-\  5.   a'  by  a". 

2.  a^  by  a"'.  4.    a  by  a-\  6.   x^  by  xK 

7.  ah^  by  ahh  10.    ?i--  by  awi 

8.  mhhymhrK  11-   «"-'' by  «-'. 

9.  a*6*  by  a"^6'.  12.   a  *    by  a~«~. 

13.  Multiply  x^y-^  -\-x^  +  x^y^  -f  x^y^  +  y^  by  x^yK 

14.  Multiply  ?/-  +  a;-y+^  -f  x-y +2  _^  ^^s^-+3  ^y  j^^-*^ 


238  TIIF.ORY  OF   EXPONENTS 

15.   Expand  {ah~^  + 1  +  a~h^)(ah~^  -  1  +  a~h^). 


First  Solution 

ah-^  + 

1      +  a~h^ 

ah-'-- 

1      +  a~h^ 

ah-^  + 

ah-^  +  a%^ 

- 

ah~^  -    1    -a~y 

+  a^&'^  +  a~h^ 

4-a"^6 

ah-^ 

+    1 
Second  Solution 

+  a~h 

(ah~ 

2  +  1  +  a~^6^)  (aH"2 

-  1  +  a~^6^) 

§  114,  =  (ah~^  +  a"^6^)2  -  12 

=  (a5&-i  +  2  a'^&o  +  a"^?>)-  1 
=  a^b-'^  +  2  +  a~^&  _  i  -  ^tft- 1  -|.  i  +  a~^b. 
Expand : 

16.  (a^  4- 6^)(a' -  6^").  20.  (x^^ -xhj'^ +  y-^)(x^ -^y'^). 

17.  (ic*H-?/3)(i6-^-?/t).  21.  (a'  +  b-'  +  ah~'^-\-l)(a^-b~^^). 

18.  (a;-^4-10)(:c-"2--l).  22.  (1  -  a;  +  ft.•2^)(a^-3  +  a;-2-f-a;-i). 

19.  (a;i-4)(a;^  +  5).  23.  (a-^  +  6"2  +  c^)(a-' +  ^'"^ +2  c^). 

Divide : 

24.  a^  by  a^  26.    a^  by  a'l  28.    x^  by  .x'i 

25.  a^  by  a^.  27.    a;^  by  x~K  29.    ic""2  by  a;"-^ 

30.  Divide  x^  +  ary  -f  ?/4  by  a;-/. 

31.  Divide  a""  +  a-2?>  +  b^  by  a-^ft. 

32.  Divide  x*-\-2  a^  +  3  aV  +  a^x  -  a*  by  aV. 


THEORY   OF   EXrONENTS  239 

33.    Divide  6"^  +  3  a"^  -  10  a  'b  by  a^-^  -  2. 
Solution 

ah-^  -  2)/>  -^  +  3  a"^  -  10  a-^b(a~^  +  5  a-^b 
fc-i  -  2  a~^ 

5  a~i  -  10  a-16 

Divide : 

34.  a  —  b  by  a^  -f-  i/^.  38.  a;  —  1  by  a; «  +  x^  -}- 1. 

35.  a  —  6  by  a^  —  6^.  39.  x^  —  2 +  x~^  hy  x^  —  x~^. 

36.  a  -f-  6  by  a^  +  bK  40.  3  —  4  x-^  +  x'^  by  «-*  —  3. 

37.  «2  +  b"^  by  a'  +  6*.  41.  a=*  -  6»  by  a^  +  6^. 

Simplify  the  following: 

42.  (ai)l  48.    (-ai)\  54.  (8-»)^ 

43.  (a-*)«.  49.    (-ay.  55.  (ICr^^. 

44.  (a-^-^.  50.    (-a^r\  56.  (-^^a^T^. 

4  3  

45.  A/xi  51.    ^<rh   '.  57.    (^a-*6*)-*. 


46 


*—  ,; 


.   A/a"^.  52.    \'.r^//-l  58.    (^m-'/i-i)^. 

4  

47.    ^ar\                     53.    Va^.  59.  (4a^y-V)2. 
Expand  by  the  binomial  formula: 

60.  (J-b^y,              62.    («-'-6«)\  64.  (a-i+.J)-''. 

61.  (ai  +  fr^.              63.    (x-^-y^y.  65.  (l-x^. 

Extract  the  square  root  of : 

66.  x^-^2x^-hSx  +  ^x^  +  3  +  2x~^  +  x-\ 

67.  jr  +  )/-{-4z-'-2xy^-\'^xz-^-4}/h-\ 

68.  a  4-  4  6*  +  9  c^  -  4  ah^  +  6  q^c^  -  12  6^cJ. 


240  THEORY   OF   EXPONENTS 

Extract  the  cube  root  of  : 

69.  a-  +  6a^  +  12a^  +  8. 

70.  a-Sah^-i-Sah^-b^ 

71.  8x-''-12x-hj-\-6xk/'-i/. 

72.  x^  —  6x-\- 15x^  —  20 -{-15  x~^^  — 6  xr^  +  x~K 

73.    Factor  4:X~^  —  9y~',  and  express  the  result  with  positive 
exponents. 

Solution.  — By  §  152,      4a;-'-^ -9?/-2  =  (2x-i  +  3  ?/-i)(2a:-i  -  3?/-i) 


\x     y)  \x     y) 


Factor,  expressing  results  with  positive  exponents : 

74.  a-^-h--.  79.  ^-x-K 

75.  ^-x-\  80.  0^  +  2  + a-'. 

76.  l(S-a-\  81.  6*- 8 +  166-*. 

77.  21-1)-^,  82.  12~x-^-x-^. 

78.  h-^  +  y-^.  83.  2-3a;-i-2x-l 

Solve  for  values  of  x  corresponding  to  principal  roots,  and 
test  each  result : 

92.  x'^  =  &. 

93.  a^~^  =  144. 

94.  25a;^  =  l. 

95.  a^2  =  243. 

96.  a^^  +  32  =  0. 

97.  x^  —  a^  =  0. 

98.  if* -64  =  0. 

99.  a;^^-27  =  0. 


84. 

0.^  =  7. 

85. 

^■t  =  8. 

86. 

x^  =  9. 

87. 

x^  =  Sl, 

88. 

4x*  =  72. 

89. 

x-^  =  12. 

90. 

ia.'^=25. 

91. 

2:^-==^ 

THEORY  OF   EXPONENTS 


241 


Simplify,  expressing  results  with  positive  exponents: 


100. 
101. 
106. 
107. 


102. 


103.  ('i^::!?L^)-^ 


104. 
105. 


27'- 
3  a^  X  4  a-» 


i 


3*x3*» 


109. 


a^j/*  X  ^a;-^  x  x 

9r+l  3r+l 


i 


108.  -57^=:^ 

Vy"*  X  xy 


110. 


m 


.i~ni 


(30^-^ 


^m*  —  Vw' 


m^  4-  7t^ 


6V^ 


112. 


113. 


114. 


115. 


116. 


18 

4 


/2a*--6*Y» _ /_1_\ 


117.  Find  the  value  of 

j(_2)-i-f-(-32)-^|-«-hJ16tx(-8)''i-l 

118.  Extract  the  square  root  of 

^  +  ^-2a;+,— ,4-x«-4V^Y"*- 

y      y-i  4y-» 

MILXE^S   STAND.    ALO. —  16 


RADICALS 


315.  An  indicated  root  of  a  number  is  called  a  radical;  the 
number  whose  root  is  required  is  called  the  radicand. 

V5a,  (0:^)3,  \/a^  +  2,  and  (x  +  y)i  are  radicals  whose  radicands  are, 
respectively,  5  a,  x^,  a^  +  2,  and  x  +  ?/. 

316.  An  expression  that  involves  a  radical,  in  any  way,  is 
called  a  radical  expression. 

317.  In  the  discussion  and  treatment  of  radicals  only  prin- 
cipal roots  will  be  considered. 

Thus,  VI6  will  be  taken  to  represent  only  the  principal  square  root  of 
16,  or  4.     The  other  square  root  will  be  denoted  by  —  VIO. 

318.  A  number  that  is,  or  may  be,  expressed  exactly  with- 
out a  root  sign  of  any  kind  is  called  a  rational  number. 

3,  I,  a— 6,  and  V25  are  rational  numbers. 

319.  A  number  that  cannot  be  expressed  exactly  without 
a  root  sign  of  some  kind  is  called  an  irrational  number. 

x^,  \/^,  1  4-  V^,  and  V  1  +  V8  are  irrational  numbers. 

320.  An  expression  is  irrational,  if  it  contains  an  irrational 
number,  otherwise  it  is  rational. 

321.  When  the  indicated  root  of  a  rational  number  carinot 
be  exactly  obtained,  the  expression  is  called  a  surd. 

\/2  is  a  surd,  since  2  is  rational  but  has  no  rational  square  root. 
V 1  +  VS  is  not  a  surd,  because  1  +  V3  is  not  rational. 

Radicals  may  be  either  rational  or  irrational,  but  surds  are 
always  irrational. 

Both  \/4  and  VS  are  radicals  but  only  VS  is  a  surd. 

242 


RADICALS  243 

322.  The  order  of  a  radical  or  of  a  surd  is  indicated  by 
tlie  index  of  the  root  or  by  the  denominator  of  the  frac- 
tional exponent. 

\n  +  X  and  (b  +  x)i  are  radicals  of  the  second  order. 

323.  A  surd  of  the  second  order  is  called  a  quadratic  surd; 
of  the  third  order,  a  cubic  surd;  and  of  the  fourth  order, 
n  biquadratic  surd. 

324.  Graphical  representation  of  a  quadratic  surd. 

In  geometry  it  is  shown  that  the  hy[)otenuse  of  a  right 
triangle  is  equal  to  the  square  root  of  the  sum  of  the  squares  of 
the  other  two  sides;  consequently,  a  quadratic  surd  may  be  rep- 
resented graphically  by  the  hyjwtenuse  of  a  right  triangle  whose 
other  two  sides  are  such  that  the  sum  of  their  squares  is  equal 
to  the  radicand. 

Thus,  to  represent  V5  graphically,  since  it  may  be  observed 
that  5  =  2*--i-l-,   draw  OA  2  units   in 
length,  then  draw  AB  1  unit  in  length  ^ 

in  a   direction    perpendicular   to   OA. 

I  )raw  OB,  completing  the  right-angled        ^^ , 

triangle  OAB. 

Then  the  length  of  OB  represents  V5  in  its  relation  to  the 
unit  length. 

It  will  be  observed  that  >/5  can  be  represented  graphically  by  a  line  of 
eract  length,  though  it  cannot  be  represented  exactly  by  decimal  figures, 
for  \  ')  =  L'.'i:;*)  •  .  .  ,  an  endless  decimal. 

EXERCISES 

325.  Represent  graphically : 

1.  V2.  3.    Vl3.  5.    V34.  7.    \^. 

2.  Vio.  4.    Vl7.  6.    V53.  8.    Vjf. 

326.  In  the  following  pages  it  will  be  a.s.suined  that  irrational  numbers 
obey  the  same  laws  as  rational  numbers.  For  proofs  of  the  generality 
of  these  laws,  the  reader  is  referred  to  the  author's  Advanced  Algebra. 


244  RADICALS 

327.  A  surd  may  contain  a  rational  factor,  that  is,  a  factor 
whose  radicand  is  a  perfect  power  of  the  same  degree  as  the 
radical.  The  rational  factor  may  be  removed  and  written  as 
the  coefficient  of  the  irrational  factor. 

In  VS  =  V4  X  2  and  v'54  =  \/27  x  2,  the  rational  factors  are  Vi  and 
\/27,  respectively  ;  that  is,  VS  =  2  •\/2  and  S/o4  =  8\/2. 

328.  A  surd  that  has  a  rational  coefficient  is  called  a  mixed 
surd. 

2\/2,  aVx^,  and  (a  —  6)  Va  +  5  are  mixed  surds, 

329.  A  surd  that  has  no  rational  coefficient  except  unity  is 
called  an  entire  surd. 

v'5,  -v/Il,  and  \/«^  +  x^  are  entire  surds. 

330.  A  radical  is  in  its  simplest  form  when  the  index  of  the 
root  is  as  small  as  possible,  and  when  the  radicand  is  integral 
and  contains  no  factor  that  is  a  perfect  power  whose  exponent 
corresponds  with  the  index  of  the  root. 

y/l  is  in  its  simplest  form;  but  Vf  is  not  in  its  simplest  form,  because 
I  is  not  integral  in  form  ;  V8  is  not  in  its  simplest  form,  because  the 
square  root  of  4,  a  factor  of  8,  may  be  extracted;  V25,  or  25^,  is  uot  in 
its  simplest  form,  because  25*  =  (S"^)^  =  5^  =  5^,  or  v'5. 

REDUCTION   OF    RADICALS 

331.  To  reduce  a  radical  to  its  simplest  form. 

EXERCISES 

1.    Reduce  V20a*'  to  its  simplest  form. 

PROCESS 

V20a«  =  V4 a^  X  5  =  ViT?  x  V5  =  2 aV5 

Explanation.  —  Since  the  highest  factor  of  20  a®  that  is  a  perfect  square 
Is  4^6^  V20a'5  is  separated  into  two  factors,  a  rational  factor  V4a'',  and 
an  irrational  factor  Vs,  that  is,  §  291,  \/20a6  =  \/4a6  x  V'5.  On  extract- 
ing the  square  root  of  4  a^  and  prefixing  the  root  to  the  irrational  factor 
as  a  coefficient,  the  result  is  2  a^v^. 


RADICALS  245 

2.    Reduce  i/—  864  to  its  simplest  form. 

PROCESS 

\/^^864  =  v^  -  216  X  4  =  ^-216  X  ^4  =  -  6\/4 

KuLE.  —  Separate  the  radical  into  two  fojctora  one  of  which  is 
its  highest  rational  factor.  Extract  the  required  root  of  the 
rational  factor^  multiply  tJie  result  by  the  coefficient,  if  any,  of  the 
'fireii  radical,  and  place  the  product  as  the  coefficient  of  the  irror 
i  tonal  factor. 

Simplify : 

3.  Vl2.  9.    Vl62.  16.    V243aV^ 

4.  V75.                 10.    Vl8a«.  16.    ^128a«6^ 

5.  \/l6.                 11.    V2Kb.  17.    (245  aV')*- 

6.  V128.                12.    V98?.  18.    (a''  +  5a*)i 

7.  ^250.                13.    V50a.  19.    Vl8a;-9. 

8.  </32.                 14.    v640.  20.    ^/af^-2x'. 

21.  Vox^-  10xy-\-5if.         22.  (3 am' -^ 6 am  +  S a)K 

fa^ 
23.   Reduce  -i/-— i  to  its  simplest  form. 
\2^ 

PROCESS 

Explanation.  —  Since  a  radical  is  not  in  its  simplest  form  when  the 
expression  under  the  radical  sign  is  fractional,  the  denominator  must  be 
removed ;  and  since  the  radical  is  of  the  second  degree,  the  denominator 
nuLst  lie  made  a  perfect  s(iuare.  The  smallest  factor  that  will  accomplish 
I  lii.s  is  2  y.  On  multiplying  the  terms  of  the  fraction  by  this  factor,  the  larg- 
est rational  factor  of  the  resulting  radical  is  found  to  be  \—^  which  is 

a  —  ^ 

equal  to  — •     Therefore,  the  irrational  factor  is  ^2  y,  and  its  coefficient 

2y^ 


246  RADICALS 


Simplify : 

24.     V|.  „^      ^  l2aF 


30.     \ 33. 


1^ 
25.    Vi.  ^    ^  "      ^3  2/^* 

"2^ 


''•     ^"^-  31.    J"^.  34.     Ji^ 

27.     VI.  ^^«'  ^'3a^2 


.3^ 


31.    \/::,;i4-.  34 

28.       V^.  ,-  ,^r— 

29.  n  ■      ^^-  i'-        ''■  V5T^ 

36.     (a +  6)./^.  37.     (^^-1^+* 

332.  Although  |-  =i,  it  does  not  follow  without  proof  that 
64^  =  64^,  for  each  fractional  exponent  denotes  a  power  of  a 
root  of  64,  and  the  roots  and  powers  taken  are  not  the  same 
for  64^  as  for  64^.  By  trial,  however,  it  is  found  that  each 
number  is  equal  to  8 ;  and  in  general  it  may  be  proved  that 

pm  p 

a«"'  =  a'';  that  is, 

A  number  having  a  fractional  exponent  is  not  changed  in  value 
by  reducing  the  fractional  exponent  to  higher  or  lower  terms. 

EXERCISES 

333.  1.    Keduce  v9  d^  to  its  simplest  form. 

PROCESS 

-^Wd'  =  VWW  =  (3  ay'  =  (3  a)*  =  ^3^ 
2.    Eeduce  V64  aW  to  its  simplest  form. 

PROCESS 


S/64  a'b''  =  </2^aW)'^  =  b(2  ab)  '  =  b  (2  ab)  •'  =  b  V4  a'b'- 
Simplify  : 
3.    a/36.  5.    a/1600.  7.    <'%Fb¥. 


^25.  6.    ^/27a'.  8.    V121aV. 


RAblCALS  247 

334.    Simplify: 
1.    \  (300.  5.    \^I89.  9.    \/U4.  13.    VJ. 


2.    V500.  6.    V84.  10.     \/8i. 

3.  </m.        7.  \/72.         11.  ^;34;i 


»■^g• 


15. 


4.    V3000.  8.    V192.  12.    V289.  \35"s 

16.  V405  ay.  18.    V8-20  6*.  20.    ^b^^. 

17.  (13oa^y)i  19.    5V4a-  +  4.  21.    (16  a; -16)*. 

_A]L.(^^,  25.    (l-ur^)jLEZ±:§. 

a;-2.y\     2y  ^^l-l-aj  +  a^ 

23.  V27c*-36c  +  12.  26.    (4  cr**- 24 a'^a; 4-36 aar^*. 

24.  </«=»- 2  ary  +  y*.  27.    (^-^"^^3/^+3  a^^y^-x^)*. 

335.    To  reduce  a  mixed  surd  to  an  entire  surd. 

EXERCISES 

1.    Express  2  a  V5  6  as  an  entire  surd. 

PROCESS 

2  a  V57;  =  VHZ*  VSft  =  V4a*x56  =  V20^ 

Rule.  —  liaise  the  coefficient  to  a  power  correftponding  to  the 
iidex  of  the  given  radical^  and  introduce  the   result  under  the 
radical  sign  as  a  factor. 

Express  as  entire  surds : 

2.  2V2.  6.   S^/S.  10.  JV2.  14.  i^/^. 

3.  3V5.  7.    4V5.  11.  fv'«\  16.  f  Vff  a*. 

4.  5V2.  8.   iV8.  12.  jV6c.  16-  |\/li. 

5.  3v^2.  9.    a^^/b.  13.  fV|.  17.  J\/3|. 

18.  '^±yJ^^.     19.   ^±iJl §-.    20.    l(a-6)i 

a*  — y^a*4-y  a       1^         a-f-4  «6 


248  RADlAvLS 

336.    To  reduce  radicals  to  the  same  order. 

EXERCISES 

1.  Reduce  V3,  V2,  and  V4  to  radicals  of  the  same  order. 

PROCESS 

^3  =  3i  =  3rV='^33^^27 
V2  =  2^-  =  2T^=  W^^v'el 

Rule.  —  Express  the  given  radicals  with  fractional  exponents 
having  a  common  denominator. 

Raise  each  number  to  the  poiver  indicated  by  the  numerator  of 
its  fractional  exponent,  arid  indicate  the  root  exp>ressed  by  the 
common  denominator. 

Reduce  to  radicals  of  the  same  order: 

2.  V2  and  ^^.  9.  ^ah,  -Vab\  and  ^2. 

3.  V5  and  a/6.  10.  Va,  Vb,  ^x,  and  ^^. 

4.  a/7  and  VlO.  11.  -\/a-\-b  and  -\/x-\-y. 

5.  a/IO,  a/2,  and  Vl.  12.  Vf,  Vj^,  and  2  V5. 

6.  a/4,  a/2,  and  V3.  13.  Va^,  Vx^,  and  V^ 

7.  A/i3,  a/5,  and  -^4.  14.  (a  +  b)  V^~^,  and  a/o^^. 

8.  a/3,  a/5,  and  a/2V-  15.  Va  +  b,  ^d-  +  6^ and  Va -  6. 

16.  Which  is  greater,  ^5  or  V2?   Vl  or  VS  ? 

17.  Which  is  greater,  ^3  or  v4?  3  a^  or  2  a^-T ? 
Arrange  in  order  of  value  : 

18.  A^S,  V2,  and  a/7.  21.    a/2,  Vl,  V2l,  and  a/4. 

19.  V2,  a/4,  and  x/5.  22.    a/7,  a/48,  a/4,  and  a/63. 

20.  a/2,  a/3,  and  ^30.  23.    a/4,a/2,a/5,  A/i3,and  a/IM). 


RADICALS  249 

ADDITION   AND    SUBTRACTION   OF   RADICALS 

337.  Radicals  that  in  their  simplest  form  are  of  the  same 
order  and  have  the  same  radicand  are  called  similar  radicals. 

riiiis,  2  VS,  a  V3,  and  7  VS  are  similar  radicals. 

338.  Pkinciple.  —  Onhf  similar  radicals  can  be  united  into 
out  terni  hy  addition  or  subtraction. 

EXERCISES 

339.  1.    Find  the  sum  of  V50,  2  \/ 8,  and  6  Vf 

PROCESS  Explanation. — To  a.scertain  whether  the  given 

expressions  are  similar  radicals,  each  may  be  re- 

\0\J^=    oV-  duced  to  its  simplest  form.     Since,  in  their  simplest 

2  \/8  =    2  V2  form,  they  are  of  the  same  degree  and  have  the  same 

/,     /T q    /o  radicand,  they  are  similar,  and  their  sum  is  obtained 

2Lr:^ jL—  by  prefixing  the  sum  of  the  coefficients  to  the  com- 

Sum  =  10  V  2  mon  radical  factor. 

Find  the  sum  of : 

2.  V50,  Vi8,  and  V98.  5.  V28,  V63,  and  V700. 

3.  V27,  Vl2,  and  V75.  6.  \/250,  -s/U,  and  ^54. 

4.  V20,  V80,  and  V45.  7.  ^128,  x/686,  arid  </j. 

8.  vTaS,  \/320,  and  \/625. 

9.  \/500,  %/108,  and  ^^^=^32. 

10.  Vi,  Vl2i,  Vi,  and  VI}. 

11.  Vi,  V75,  I V3,  and  Vl2. 

12.  Vf,  iV3,  J  V9,  and  Vl47. 

13.  V40,  V28,  ^^25,  and  Vl75. 

14.  V375,  V44,  Vr92,  and  V99. 


250  RADICALS 

Simplify : 

15.    V245  -  V405  +  V45.      23.    Vl2S^  + -^WTKx  -  ^/W 


16.  V12  +  3V75-2V27. 

17.  5V72+3V18- V50. 


/a  /a  /a 

17.    5V72+3V18-V5'0.  '    ^x'^\'~^?' 


19.    V112-V343+V448.  ^^^'       ^    ^.'/         ^  ^'f 


-  VfS-V^^W^ 


20.  ^135 -v'625  + -2/320. 

21.  -^|  +  ^T  +  ^5|.  ^fftc  ■    "^ac  ■    V„i 


-  xfeW^NE- 


22.    V864-V4000  +  V32.     27.    V(a  +  byc- ^(a-bfc. 
28.    6^|f  +  4^If-8v'|||. 


5/ 7T7^ — 3    .    r.  5/7r— 7  3/1::^ —    ,      3/ 


29.    V-96a;^  +  2V3a;4-V5a;  +  V40x4. 


30.    ■</abx--\/a'b'a^  +  </'Sa'b'x\ 


31.    V3ar^  +  30x-  +  75a;-V3ar^-6x2  +  3a;. 


32.  V5  a'  +  30  a^  +  45  a^  -Voa'-  40  a^  -f-  80  a^ 

33.  V50  +  ^9  -  4  V|  +  -s/^^^tM:  +  v  27  -  a/64. 

34.  V|+6V|-iV18+v'36--v^I|  +  -^125-2VA-     ' 

35.  (f)^-(|)-^+V(S)^+Vi:35-</(T|p. 

36.  5  .  2-*+  2-^+3  .  2-^+  3  •  5-^  •  2^^  +  ^^ff^. 

MULTIPLICATION  OF  RADICALS 

340.  a^  xa^  =  a^-^^  =  a^+^  =  a^. 

That  is,  Va  x  V «  =  Va^  x  V<x"  =  Va'^ 

Since  fractional  exponents  to  be  united  by  addition  must  be 
expressed  with  a  common  denominator,  radicals  to  be  united  by 
multiplication  must  be  expressed  with  a  common  root  index. 


RADICALS  251 

EXERCISES 

341.   PROCESSES.  —  1.    V7  X  Vi>  =  V35. 

2.  5V3x2\/l5  =  10V45  =  10x3V6  =  30>/6. 

3.  2V^x3\/i>  =  2v^'27x3v/4  =  6\/I()8. 

Rule.  —  If  the  radicals  are  not  of  the  same  order ^  reduce  them 
to  the  name  order. 

Multiply  the  coefficients  for  the  coefficient  of  the  product  and  tJie 
radicands  for  the  radicid  factor  of  the  product;  simplify  the  re- 
stdf,  if  necessary. 

Multiply: 

4.  V2  by  V8.  11.   2\/6  by  3V0. 

5.  V2  by  V6.  12.   3  V3  by  2\/5. 

6.  V3  by  Vi5.  13.    \/5  by  VlO. 

7.  2V5by3ViO.  14.    2v250by  V2. 

8.  3 V20  by  2V2.  15.   2\/24  by  ^l8. 


9.    V2  by  3V3.  16.   2 V2  by  V512. 

10.   2^3by3^J^.  17.    ^/2xyhyS^/^. 


\/24  by  ^' 

3/ 


Find  the  value  of: 

18.    V?/iH  X  -^mhi  X  y/mu*. 
19    V2  axy  x  "s/xy  x  y/a^xy, 

20.  Vx^  X  y/x-yx  Vx-y, 


21.    Va  ~  6  X  Va*6*  x  V(a-6)-«. 

22.  VlxV|xV|.  26.    -V^i  X  v/f  X  VI. 

23.  v.lxV|xV|.  26.    16*  X  2^x32^. 

24.  </|  X  \/|  X  Vl.  27.   27'  X  9*  X  8li. 


252  RADICALS 

28.    Multiply  2  V2  +  3  V3  by  5  V2  -  2  V3. 
Solution 

2V2  +  3V3 
5V2-2V8 
20     +  15  V6 

-    4V6-18 


20      +11V6-18=2+  llv^e. 
Multiply : 

29.  V5  +  V3  by  V5  -  V3. 

30.  V7  +  V2  by  V7  -  V2. 

31.  V6  -  V5  by  V6  -  VS. 

32.  5  -  V5  by  1  +  V5. 

33.  4V7  4- 1  by  4V7  -  1. 

34.  2V2  +  V3  by  4 V2  +  V3. 

35.  2 V3  +  3V5  by  3V3  +  2V5. 

36.  3  a  +  V5  by  2  a  —  V5. 

37.  2V6  -  3 V5  by  4V3  -  VlO. 

38.  a^  -  a6V2  +  b''  by  a^  +  abV2  +  &l 

39.  X  —  ViC2/2;  +  2/2;  by  Vx  +  V?/^;. 

40.  x^x  —  x-\/y  +  2/ Va;  —  y^y  by  Va;  +  V2/. 
Expand : 


41.    (\3  +  V5)(\3-V5).       43.    (Al6  +  Vll)(\'6-Vll). 


42.  (aJ9  +  V6)(\'9-V6).        44.    (\oa  +  aV5)(\5a-aV5; 
45.  (\7  c  +  -V5^){^7  c -  V5^^). 


46.    (\'l4 oj  +  a;V27)(\l4  x  -  x V2< 


RADICALS  253 

DIVISION  OF  RADICALS 

342.  ai  -f-  a^  =  a^  ^  =  a^~^  =  aK 

That  is,  Va  -5-  ^/a  =  Va"'  -f-  V a-  =  V^^ -^  «'  =  Va. 

In  division,  when  one  fractional  exponent  is  subtracted  from 
another,  the  exponents  must  be  expressed  with  a  common  de- 
nominator. When  one  radical  is  divided  by  another,  the 
radicals  must  be  expressed  with  a  common  root  index. 

EXERCISES 

343.  Processes.  —  1.   V(iO  h- Vi2  =  V5. 

RiTLE.  —  If  necessary f  reduce  the  radicals  to  tJie  same  order. 

To  the  quotient  of  the  coefficients  annex  the  (jnotient  of  the 
radicands  written  under  the  common  radical  sign^  and  reduce 
the  result  to  its  simplest  form. 

Find  quotients : 

4.  VSO-hVS.  12.  2S!/12--V8. 

5.  V72-i-2V6.  13.  Va^^Vxy. 

6.  4V5  +  V40.  14.  \/2aF -i- ^a*6\ 

7.  6V7-i-Vl26.  15.  ^7tV--V2^. 

8.  ^-j-V2.  16.  ^9^^^-4-V3^. 

9.  7\/135h-%/^.  17.  </r^-f-\/2^ 

10.  7V75H-5V28.  18.    Va  —  6  -t-  Va  +  6. 

11.  vl6 -5-^/32.  19.   3^1 -^V|. 


254  RADICALS 

20.  Divide  VIE  -  V3  by  V3. 

21.  Divide  V6  -  2 V3  +  4  by  V2. 

22.  Divide  V2  -f-  2  +  iV42  by  i  V6. 

23.  Divide  5  V2  -}-  5  V3  by  VlO  +  Vl5. 

24.  Divide  5  +  5 V30  +  36  by  V5  -f  2 V6. 

INVOLUTION    AND    EVOLUTION   OF   RADICALS 

344.  In  finding  powers  and  roots  of  radicals,  it  is  frequently 
convenient  to  use  fractional  exponents. 

EXERCISES 

345.  1.  Find  the  cube  of  2Va^. 

Solution.       (2 \^^^Y  =  2\a;^x^^  =  8  aV'  =  SVo^^  ^  g  ax^Vax. 

2.  Find  the  square  of  3^a^. 

Solution.       (.3  y/xpy^  =  9(x^)2  =  dx^  =  9  Va^  =  9  x\/x^. 

3.  Obtain  by  involution  the  cube  of  ■y/2  -f- 1. 

Solution 

(V2  +  1)3  =  (  V2)3  +  .3(\/2)2  .  1  +  3V2  .  12  +  1-3 
.     :3  2V2  +  6  +  3V2  +  1 
=  7  +  5V2. 
In  such  cases  expand  by  the  binomial  formula. 


Square : 

Cube: 

Involve  as  indicated : 

4.    SVab. 

9.    2Vo. 

14.    (-2V2a6/. 

5.    2  a/3  a;. 

10.   3V2. 

15.    {-■V2</~xf. 

6.    xi/2x^. 

11.   2v'^. 

16.    (-V2\/'aary. 

7.    7iV4l>. 

12.    Va'h\ 

17.    (- 2  Va^^?//'. 

8.    aV^. 

13.    ^4  7i^ 

n     n 

18.    (-ocrx-'^/. 

RADICALS  255 

Expand : 

19.  (2-hV(3)=^.         22.    (2-V3)»  25.    (V;c  ±  1)». 

20.  (2  +  V2)'-         23.    (VT-VG)'.  26.    (Va-V6)« 

21.  (2  4-V5/.         24.    (2V2-V3)l         27.    (V«  ±  1)^ 

28.  What  is  the  fourth  root  of  V2x  ? 
Solution.  -J^v^  =  [(2a:)i]^  =  (2x)*  =  v^. 

Find  the  square  root  of :  Find  the  cube  root  of  : 

29.  V2.  32.    \/x*.  35.    V2a;.  38.    -21 V^. 

30.  \/5.  33.    \/x^.  36.    VTV.         39.    -  Vo^. 

31.  </^.         34.    -v/^r^.  37.    -s/W^HM.     40.    -  64</^. 
Simplify  the  following  indicated  roots : 


41.   ^iVU^.  43.    (V8a«a^*.  /p^_ 

42.  -^v^:  44.  (v^)^.         '  \WyJ 

Rationalization 
346.   Suppose  that  it  is  required  to  find  the  approximate 

value  of  — T=»  having  given  V3  =  1.732  •••. 
V  o 

1.732  ...|1.000000|. 577...  3)1.732  ..» 

8660  .577... 

We  may  obtain  a  decimal  approximately  equal  to  — -y  as  in 

V3 
the  first  process  (incomplete),  by  dividing  1  by  1.732.-.;  but 
a  great  saving  of  labor  may  be  effected  by  first  changing  the 
fraction  to  an  equivalent  fraction  having  a  rational  denomi- 
nator, thus :  ^  ^      1.V3  ^  V3 

V3~V3.V3"   3  ' 

and  employing  the  second  process. 


266  RADICALS 

347.  The  process  of  multiplying  a  surd  expression  by  any 
number  that  will  make  the  product  rational  is  called  rationali- 
zation. 


The  factor  by  which  a  surd  expression  is  multiplied  to 
render  the  product  rational  is  called  the  rationalizing  factor. 

349.  The  process  of  reducing  a  fraction  having  an  irrational 
denominator  to  an  equal  fraction  having  a  rational  denomi- 
nator is  called  rationalizing  the  denominator. 

EXERCISES 

350.  Find  the  value  of  each  of  the  following  to  the  near- 
est fifth  decimal  place,  taking  V2  =  1.41421,  V3  =  1.73205, 
and  V5  =  2.23607  : 

1.  A.  3.    J_.  5     J^. 

V2       '  "    V8  *    VoO 

2.  -^.  4.     J^.  6.         1 


V5  V45  V125 

Rationalize  the  denominator  of  each  of  the  following,  using 
the  smallest,  or  lowest,  rationalizing  factor  possible : 

Vaj'  ■    Vl2  '    Va-b 


ax  ^  -       Va 


8-     -7==-  10-     ^^-  12.    Jl 


■^^ 


V2a'x  Vax"  *    \        x-^2 

351.  A  binomial,  one  or  both  of  whose  terms  are  surds,  is 
called  a  binomial  surd. 

y/2  +  V5,  2  +  \/5,  \/2  +  1,  and  VS  -  ^^2  are  binomial  surds. 

352.  A  binomial  surd  whose  surd  or  surds  are  of  the  second 
order  is  called  a  binomial  quadratic  surd. 

V2  +  VS  and  2  +  V5  are  binomial  quadratic  surds. 

353.  Two  binomial  quadratic  surds  that  differ  only  in  the 
sign  of  one  of  the  terms  are  called  conjugate  surds. 

3+  VS  and  3—  \/5  are  conjugate  surds  ;  also  VH-\-  V2  and  VS—  V2. 


RADICALS  267 

354.  The  product  of  any  two  conjugate  surds  is  rational. 
For,  by  §  114,    (Va  +  y/b)(\/a  -  Vb)  =  ( Va)'^  -  (  V6)2  =  a  -  b. 

Hence,  a  binomial  quadratic  surd  may  be  rationalized  by  multi- 
plying it  by  its  conjugate. 

EXERCISES 

2 

355.  1.    Rationalize  the  denominator  of  -' 

3- V6 

Solution 

2       ^        2(3  4- V5)         _^2(3+>/o)_3-hV6^ 
3_V5      (3-V5)(3  +  >/6)    '      »-6  2 

2.    Rationalize  the  denominator  of    - — ~     _  » 

V7  +  V3 

Solution 

\/7-V3^(V7-V3)(V7->/3)^7-2V21-t-3^5-V2r 
V7+V8      (>/7+V3)(V7-V3)  '-8  2       ' 

Rationalize  the  denominator  of : 

3  3  5     V|i^>L?.  7     a-2V6 

2  +  V3'  '    V3-V2  '    a  +  2V6* 

4.  5  g    5-3V2  g     Vg-hVA^ 

\/5-V3*  *    2-V2   '  *    V«-Viy 

g     4V|+6V3^  jj     Va'  +  g-i-l-l 

;W3~2V2  '     >/a*Ta+Wl 

10.    x-y/^-^,  12.     Va?  +  .V-Vg-V. 

a?  +  Vaj*--l  \/'x-\-y-^\^x  —  y 

Reduce  to  a  decimal,  to  the  nearest  thousandth : 

13.  811^.  14.  _1_.  16.  :vl±V2. 

2-V3  3-f\/5  V3-V2 

mh-nk'*  «'t^ni).    mo.  — 17 


258  RADICALS 

16.  Rationalize  the  denominator  of      _  ~  ^  _  ~"  ^_. 

V2  -f-  V3  4-  Vo 
Solution 

v^  -  V3  -  V5  ^  ( V2  -  V5)  -  \/3  ^^  (  V2  -  Vd)  +  VS 

\/2  +  V3  +  \/5      ( V2  +  V3)  +  V5      (>/2  +  V3)  -  V5 

^2-2  VlO  +  5-8^4-2  VfO 

2  +  2  V6  +  3  -  o  2  V6 

^  2  -  ViO  ^  2  V6  -  2  Vis  ^  V6  -  \/T6       • 
V6  6  3 

Rationalize  the  denominator  of: 

17.  V2-V^W7.  ,,^       ^_        V         .- 
V2  +  Vo  +  V7  V2  +  V3  4-  V5 

18.  V3+V2  20.    2V2-3V3f4V5 
V3  +  V2  — V6  V^+V3— V5 

21.    Rationalize  the  denominator  of  —^ ^7=?  or  — -. 

Va  -h  \/6-         a^  +  ftt 

Solution.  —  By  §  134,  Va  +  Vd^,  or  a2  +  fe^,  is  exactly  contained  in 

the  sum  of  any  like  odd  powers  of  a^  and  6^,  and  also  in  the  difference 

of  any  like  even  powers  of  a^  and  b^.    The  lowest  like  powers  of  a^  and 

63  that  are  rational  numbers  are  the  sixth  powers,  which  are  even  powers. 

1         2 
Hence,  the  rational   expression    of   lowest  degree  in   which   a^  +  &3  is 

exactly  contained  is  (a^y  —  (h~sy,  or  a^  —  &*. 

Dividing  a^  —  &*  by  a^  +  &3,  the  rationalizing  factor  for  the  denomina- 
tor is  found  to  be  as  —  aV)t  +  a'^b^  —  ab^  +  a'^bs  —b  ^ . 

Multiplying  both  terms  of  the  given  fraction  by  this  factor, 

a  or      ^        =  ^(^  -  ^^^^  +  ^^^^  ~  ^^'"^  +  ^^^^  -  ^^"^^ 

Va+V/7^'      ai  +  7;l  «'-^* 

Rationalize  the  denominator  of : 

22  "^^  24  "^^  26      _^^'-. 

Va-^6  ■     ^a'-V9'  '    ^a--\/x 

23.      ^_^      „.  25.     /.^  +  ^.  27.     ^  J^^. 

Vx  4-  Vi/  V  a  —  V^  V  ic+  V?/^ 


RADICALS  269 

Square  Root  of  a  Binomial  Quadratic  Surd 

356.    To  find  the  square  root  by  inspection. 

The  square  of  a  binomial  may  be  written  in  the  form 

(a  -I-  by  =  (a-  +  b-)  -\-2ab. 

Thus,       ( V2  +  Vr))2  =  (L>  +  (>)  +  2  Vi2  =  8  +  2  Vl2. 

Therefore,  tlie  terms  of  the  square  root  of  8  +  2  Vl2  may  be 
nhtained  by  separating  Vl2  into  two  factors  such  that  the 
sum  of  their  squares  is  8.     They  are  ■\/2  and  V6. 


V8-f2Vl2=  V2+ V6. 


Principle.  —  Tlie  terms  of  the  square  root  of  a  binomial 
'/"odratic  surd  that  is  a  jterfect  square  may  be  obtained  by  divid- 
in/j  the  iri'ational  term  by  2  and  then  separating  the  quotient  into 
tiro  factors,  the  sum  of  whose  squnres  is  the  rational  tei-m. 

EXERCISES 

357.    1.    Find  the  square  root  of  14  +  8  V3. 
Solution 
14  +  8  V3  =  14  -f-  2  (4  VS)  =  14  +  2  >/48. 
Smce  \/48  =  V6  x  VS  and  14  =  6  +  8, 


Vl4  +  8  V3  =  Ve  +  \/8  =  Ve  +  2  V2. 

2.    Find  the  square  root  of  11  —  0  ^2. 
Solution 


Vn-6V^  =  Vll  -  2  Vl8  =  \/9  -  v'2  =  3  -  \/2. 

Find  the  square  root  of : 

3.  12  +  2V35.  7.    11  +  2V30.  11.  12-h4V5. 

4.  16-2V60.  8.    7-2ViO.  12.  11+4V7. 

5.  15-f2V26.  9.    12-(>V.S.  13.  15-6\/6. 

6.  16-2V55.  10.    17-hl2V2.  14.  18  +  6V0. 


260  RADICALS 

Find  the  square  root  of : 

15.  3-2V2.  17.    a'-\-b-\-2aVb. 

16.  6  +  2 V5.  18.    2a-2Va2-62. 

358.  To  find  the  square  root  by  using  conjugate  relations. 

This  method  is  useful  in  the  more  difficult  cases.  It  depends 
upon  the  following  properties  of  quadratic  surds. 

359.  Principle  1. —  The  square  root  of  a  rational  number 
cannot  be  partly  rational  and  partly  a  quadratic  surd. 

For,  if  possible,  let  Vii/  =  Vh±m,  Vy  and  yfh  being  surds. 

By  squaring,  2/  =  6  ±  2  m  V6  +  m^, 

and  V^^±y-^^-&^ 

2  m 

which  is  impossible,  because  (§  321)  a  surd  cannot  be  equal  to  a 
rational  number. 

Therefore,  V?/ cannot  be  equal  to  V6  ±  m. 

360.  Principle  2.  —  In  any  equation  corUaining  rational 
numbers  and  quadratic  sards,  as  a-{-'\/b  =  x-\-^y,  the  rational 
parts  are  equal,  and  also  the  irrational  parts. 

Given  a-\-\/b~x  +  Vy.  (1) 

Since  a  and  x  are  both  rational,  if  possible,  let 

a=x  ±in.  (2) 

Then,  x ±m  +  Vb  =  x  •}■  Vy,  (•>) 

and  Vy  =  y/h±  m.  (4) 

Since,  §359,  equation  (4)  is  impossible,  a  =  x±m  is  impossible; 
that  is,  a  is  neither  greater  nor  less  than  x. 

Therefore,  a  =  x,  and  from  (1),  Vh  =  Vy. 

Hence,  if  a  -j-  Vb  ~  x  -h  Vy.  a  —  x  ynd  V^  =  Vy. 


RADICALS  261 

361.  Pkinciple  .*^. —  1/  a-\-^b  arid  a  —  y/b   are   binomial 

(juadratic   .siirdj<    and   \a 4- V6  =  Vx -h Vy,   then    \tt— V6  = 
Va;  — Vy. 

To  exclude  imai^iiiHrv  numbers  from  the  duicussion,  suppoHe  that 
'/  -Vbiti  positive. 

Given  V a  +  \/6  =  y/x  -{■  Vy. 

Squaring,  §  277.  a  +  Vfe  =  x  +  2  Vxy  +  //• 

Therefore,  §  :i«0,  a  =  x  +  y  and  V6  =  2  y/xy ; 

whence,  Ax.  2,  a  —  Vft=  x  -\-  y  —  2  Vxy. 

Hence,  §  289,  Va  -  \/6  =  v^  -  v^. 

EXERCISES 

362.  1.    Find  the  square  root  of  21  -f  6  VTO. 

Solution 


Let  y/x  +  y/'y  =  V2I  -f  6  VlO.  (1) 

Then,  §  361,  ^x-y/y=  V2I  -  6  VIO.  (2) 

Multiplying  (1)  by  (2),  x-y=  V441  -  360  =  VSl, 

or  x-y  =  9.  (3) 

Squaring  (1),  §  277,    x  +  2  y/xy  +  y  =  21  +  tt  VTo. 
Therefore,  §  300,  x  +  y  =  21.  (4) 

Solving  (4)  and  (3),  x  =  15,  y  =  6. 

.-.  y/x  =  v/l6,  y/y  =  y/6. 


Hence,  from  (1),  V2I  +  6  VlO  =  vTs  +  >/6. 

Find  the  square  root  of : 

2.  25-hlOV«.  8.    1H-I-6V7.  14.  2-I-V3. 

3.  1<)4-<>V2.  9.    21-8V6.  15.  6  +  V35. 

4.  45  +  30 V2.  10.   47-I2VII.  16.  l-fiV2. 

5.  8o-14\/G.  11.   064-32V3.  17.  2-+-^V6. 

6.  ll-f6V2.  12.   35-12V(i.  18.  .S0H-20V2. 

7.  24-8V0.  13.    56-12V.'3.  19.  I8-6V5. 


2Gl^  radicals 

RADICAL  EQUATIONS 

363.  An  equation  involving  an  irrational  root  of  an  un- 
known number  is  called  an  irrational,  or  radical,  equation. 

X-  =  S,  Vx  +  1  =  Vx  -  4  +  1,  and  v^x  —  i  =  2  are  radical  equations. 

364.  A  radical  equation  may  be  freed  of  radicals,  wholly  or 
in  part,  by  raising  both  members,  suitably  prepared,  to  the  same 
power.  If  the  given  equation  contains  more  than  one  radical, 
involution  may  have  to  be  repeated. 

When  the  following  equations  have  been  freed  of  radicals, 
the  resulting  equations  will  be  found  to  be  simple  equations. 
Other  varieties  of  radical  equations  are  treated  subsequently. 

EXERCISES 

365.  1.    Given  V2^  +  4  =  10,  to  find  the  value  of  x. 

SOLUTIOX 

\/2x"+4=10. 
Transposing,  V'Jx  =  6. 

Squaring,  2x  =  36. 

.-.  x  =  18. 

Verification.  —  Substituting  18  for  x  in  the  given  equation  and  (§  317) 
considering  only  the  positive  value  of  \/2~x,  we  have  V3()  -f  4  =  10  ;  that 
is,  10  .=  10,  an  identity  ;  hence,  the  equation  is  satisfied  for  x  =  18. 


2.    Given  ^x  —  7  +  Vx  =  7,  to  find  the  value  of  x. 

Solution 
Vx-  7  +  Vx  =  7. 
Transposing,  Vx  —  7  =  7  —  Vx. 

Squaring,  x  —  7  =  49  -  14  Vx  +  x. 

Transposing  and  combining,     14  Vx  =  66. 
Dividing  by  14,  Vx  =  4. 

Squaring,  x  =  16. 

Verification.     V16  -  7  +  VTo  =  V9+  Vi6=  3  +  4=7:  that  is,  7  =  7. 


RADICALS  268 


3.    Given  \14  -f  Vl  -f  Va;-f8  =  4,  to  find  the  value  of  ar. 
Solution 


\l4+Vn-Va;  +  8  =  4. 


Squaring,  14  +  Vl  +  Vac  +  8  =  16. 


Transposing,  etc.,  Vl  +  Vx  +  8  =  16  -  14  =  2. 

Squaring,  1  +  Vx  +  8  =  4. 

Transposing,  etc.,  y/x-{-  8  =  4—1  =3. 

Squaring,  x  +  8  =  9. 

.-.a;  =  9 -8  =  1. 


Vebification.     Vi4  +  Vl  +  Vl  +  8=  Vl4  -f-  Vl  +3' 


=  V14  +  2  =  4  ;  that  is,  4=4. 

General  Directions.  —  Transpose  so  that  the  radical  term,  if 
tJiore  is  but  one,  or  the  most  complex  radical  term,  if  there  is 
more  than  one,  may  constitute  one  member  of  the  equation. 

Then  raise  each  member  to  a  power  corresponding  to  the  order 
of  that  radical  and  simplify. 

If  the  equation  is  not  freed  of  radicals  by  the  first  involution, 
j'l'oceed  again  as  at  first. 

Solve,  and  verify  each  result : 
4.    Vx-f  11  =4.  11.    \  -\-2\x  =  7  -Vx. 


5.    Vx  +  o  =  31.  12     V  ./•  t  1<)- \  .'-  =  2. 


6.  Vac  — a*  =  6.  13.  V2  J^— v2  ./  -  15  =  1. 

7.  \/a;-l  =  2.  14.  V^T^TT  =  2-a;. 

8.  Vsr^'c^^a.  16.  .3Vi?^^^=  3  a;  -  3. 

9.  ^/x-^b  =  a.  16.  Vx-}-2  =  Var-f  32. 


10.    1  -f  Va;  =  5.  17.    5  -  v'ir  +  5  =  Va;. 


264  RADICALS 

Solve,  and  verify  each  result : 


18.    ^x^  —  ox  +  7  -^2  =  x.         19.    V9a;-}-8  4- V9a;  — 4  =  0. 
20.    4  -  Vi^^~x^9'x^  =  S  X. 


21.    ^/2{l-x){Z-2x)-l  =  2x. 


22.    V2a?  — 1  +  V2a;  +  4  =  5. 


23.    V3a;-5  + V3ic  +  7  =  6. 


24.    Vl6a;  +  3+ Vl6a;  +  8  =  5. 


25.    A/i  -f  it'vV  +  12  =  1  4-  a;. 


26.  A7  +  3V5a;-16-4  =  0. 

27.  2x-V4ar^- V16^-7  =  1. 


28.    2Va;- V4a;- 22- V2  =  0. 


29.    V2(a;  +  1)  + V2i»-1  =  V8a;4-1. 


30.    V3  a:  +  7  +  V4  ic  —  3  =  V4  a;  +  4  +  V3  x. 


31.    V\V2a;  +  56  =  2. 


V7  +Vi 


32.    A/7  +  \l+\4+A^l4-2Va-  =  3. 

5 

33.  Solve  the  equation =  V3  ic  4-  2  +  V3  x  —  1. 

V3  i»  +  2 

Suggestion.  — Clear  the  equation  of  fractions. 

34.  Solve  the  equation  ^/^±^P-  =  ^^±1 . 

V3  a;  +  5       V3  a;  + 1 
Suggestion.  —  Some  labor  may  be  saved  by  reducing  each  fraction  to  a 
mixed  number  and  simplifying  before  clearing  of  fractions. 

Thus,  1  +  -^-—  =  1  -f  —J-  - . 

Vo  X  +  5  V  3  a;  +  1 

Canceling,  and  dividing  both  members  by  5, 
2  1 


V8  x  +  5      a/3^  +  1 


RADICALS  265 


Solve  and  verify : 


^^      V2x-f9      V2a;-f20       ._      \/2r-\-6  _V2r  +  2 
oo.     — =  — •      'kyj. 


36      V^-f  18^      32       ^  ^       ^j      Vlln-f  Vgn4-3^8 
V«+2        Vx4-6        *  *     Vir^-V2n+3      ^^ 


37      Vr^^Vs-S  ^2     2V2a;4-4^3Vj;  +  l-f  9 

V«  +  5      V»^^  '    2V2«-4     3V^Tr-9 


38.     >^=Vv-8.  ^3 

Vv  —  1      Vv  —  5 


39.    ^^^=VLzi.  44. 


Vm  4- 1  —  Vm  - 

-l 

1 

Vm4-1  + Vm- 
V4z^-34-2V2■ 

^ 
-^ 

2 
=  5. 

v«+i   vr^  V4Z+3-2V2-1 


^^      A/V5a;-9      >/V5a--21 

\V5x  +  ll     VV5^-16 
Suggest  ION.  —  First  square  both  members. 

46.  ^-3     ^V^+V3^2V3. 
Vx- V3  2 

Suggestion.  —  Begin  by  simplifying  the  first  member. 

,_        , 3 

47.  V2a?-V2aj- 7  =  -;====• 

V2ar  — 7 

48.  Solve  V^T^  +  V^^  ^  2  +  ^?HZ  for  x. 

Vx  -ho—  Vx  —  a  o 

Suggestion.  —  Rationalize  the  denominator  of  the  first  fraction. 

Solve  for  «,  and  verify : 


49.  Vx-h  Vx-(a-6)*  =  a  +  6. 

50.  aVx-6Vx  =  a*-|-6*  — 2a6. 
61.    V6 ax  —  9 a*  +  a  =  V5ax. 


6a 


62.    Vy4-3a=       ^"  -    -  Vx 

VxH-3a 


266  RADICALS 

53.    Solve  ■\'x  -h  V2  «;  H-  V3  x  =  Va  for  x. 
Solution 


Vx  +  v'2  X  + 

V3 'i  =  Va. 

(l) 

Factoring,     Vx{\  -\-  V2  + 

V3)  =  Va. 

•.  Vx=  -   --  ^«-^- 
1  +  ^'2  +  V3 

Va(l+V2-V3) 

(2) 

(1  +  V2  +  V8)(l  +  V2  -  V3) 

_  v/a(l  4-  V2  -  V3) 

2V2 

(3) 

Squaring, 

a;  =  ^(l  +  V2-  V.3)-^. 
8 

(4; 

Solve  for  x: 

54.  V2  X  -I-  V3  a:  +  V5  a;  =  V  m. 

55.  V'2  it* -|- V3  X  —  Vo  it*  =  Vc. 


56.    V^"^^^  +  V2(2--rt)  =  \3x  +  aV2. 

366.  From  §§  364,  365,  the  student  will  have  observed  that 
radical  equations  are  freed  of  radicals  either  by  rationalization 
or  by  involutiott. 

Thus,  V2a^-6  =  ()     (1)         V2^  +  6  =  0     (2) 

Multiplying  ^y         V2  x  +  G  a  2  x  —  6 

2a.'-36  =  0  2a;-36  =  0 

.-.a;  =  18  .•..!•  =  18 

If  the  positive,  or  principal,  square  root  of  2  x  is  taken, 
X  =  1S  satisfies  (1)  but  not  (2)  ;  if  the  negative  square  root 
of  2  a;  is  taken,  ic  =  18  satisfies  (2)  but  not  (1). 

It  has  been  agreed,  however,  that  the  sign  V  shall  denote 
only  principal  roots  in  this  chapter,  and  because  of  this  arbi- 
trary convention,  our  conclusion  must  be  that  (1)  has  the  root 
.r  =  18  and  that  (2)  has  no  root,  or  is  impossible. 


RADICALS  267 

According  to  this  view,  when  both  members  of  (^1)  are  mul- 
tiplied by  V2«  +  6,  no  root  is  introduced  because  V2 a? -f  6  =  0 
lias  no  root;  but  when  both  members  of  (2),  which  has  no 
root,  are  multiplied  by  V2  a;  —  6,  the  root  of  V2^  — 6  =  0, 
which  is  x  =  18,  is  introduced  (§  230). 

A  root  may  be  introduced  in  this  way  by  rationalization,  or 
i)y  the  equivalent  process  of  squaring. 

Thus,  V2^-f-6  =  0.  (2) 

Transposing,  \/2~x  =  —  6. 

Squaring,  §  277,  2  x  = .%. 

.-.  x=\K 
Verifying,  V2^1lS  -f6=6-f6=?fc0. 

The  symbol  =^  is  read  *■  is  not  equal  to.' 

EXERCISES 

367.   1.   Solve,  if  possible,  the  equation  V*— 7  —  V«  =  7. 
Solution.  —  Transposing,  squaring,  simplifying,  etc., 

Vx  =  -i. 
Squaring,  jr  =  16. 

Verification.         V16  -  7  -  VUi  =  vT)  -  VW)  =  .^  -  4  :jfc  7. 
Hence,  the  equation  has  no  root,  or  is  impossible. 

Solve,  and  verify  to  discover  which  of  the  following  equa 
tions  are  impossible ;  then  change  these  to  true  equations  : 


2.  v2xH- V2aj-3  =  1.        5.    VIx  +  l-2y/x- 1  =9. 

3.  V3  a;  +  7  -|-  V3^=  7.        6.    VTx  -  Vir  =  y/9x  —  32. 

4.  2Vi+V4a:-ll  =  l.       7.    V5a; -1  -  1  =  V6a;  +  16. 


8.    Vaj  +  1  -f  V«-i-2  -  V4  X  +  5  =  0. 


9.    V2(x*-|-3a;-6)=(aj-h2)V2. 


vx-4      V«T8  Vl9aj-V2aj-hll 


IMAGINARY    NUJMBERS 


368.  Our  number  system  now  comprises  natural  numbers, 
1,  2,  3,  ... ;  fractions,  arising  from  the  indicated  division  of  one 
natural  number  by  another;  negative  numbers  (denoting  oppo- 
sition to  positive  numbers),  arising  from  the  subtraction  of  a 
number  from  a  less  number;  surds,  arising  from  the  attempt  to 
extract  a  root  of  a  number  that  is  not  a  perfect  power;  and 
finally  imaginary  numbers,  arising  from  the  attempt  to  extract 
an  even  root  of  a  negative  number  (§  285). 

In  this  chapter  only  imaginary  numbers  of  the  second  order 
will  be  treated. 

Before  the  introduction  of  imaginary  numbers,  the  only 
numbers  known  were  those  whose  squares  are  })ositive,  now 
called  real  numbers  to  distinguish  them  from  imaginary  num- 
bers, whose  squares  are  negative. 

369.  Since  the  square  of  an  imaginary  number  is  negative, 
imaginary  numbers  present  an  apparent  exception,  in  regard  to 
signs,  to  the  distributive  law  for  evolution.     Apparently 

V"-^  X  V^^  would  equal  V(—  1)(—  1)  =  V+1  =  ±  1. 

But  by  the  definition  of  a  root,  the  square  of  the  square  root 
of  a  number  is  the  number  itself. 

Hence,     V^H!  x  V^^H:  =(V^=ri)2=  _  i^  not  -f  1-         (A) 

In  this  chapter  it  will  be  assumed  that  imaginary  numbers 
obey  the  same  laws  as  real  numbers,  the  signs  being  deter- 
mined by  (A),  which  we  call  the  fundamental  property  of 
imaginaries. 

268 


IMAGINARY   xXUMBERS  269 

370.    Powers  of  v  —  1. 

( v:=l)«  =  ( V=l)(  V^^i)  =  - 1 ; 

(V^^)«  =  (V^i)''  V-^    =  (-  l)V^=^  =  -  V^^; 
(V^=l)*  =  (V^^(V^-^=(.-l)(-l)   =-hl; 

and  so  on.     Hence,  if  /<  =  0  or  a  positive  integer, 


(B) 

( V^ri)4n+s  =  _  V- 1 ;  ( v^rT)*.+4  =4.1.' 

Hence,  any  even  power  of  y/  —  1  is  real  and  any  odd  power  w 
imaginary. 

For  brevity  V^^l  is  often  written  i. 

371.  Operations  involving  imaginary  numbers. 

BXBRCISBS 

Find  the  value  of : 

1.  (V^)'.        3.    (V^T)»«.        6.    (V^^)»«.        7.    {-if. 

2.  (V^^       4.    (V^HT)-'.       6.    (V^=T)»*.       8.    (-if. 


9.   Add  V-  a*  and  V-  16  a^ 
Solution 
vCT^i  ^  V-16a*  =  a«>/^  +  4  a=»\/^=n  =  5  a'V^. 

Simplify : 


10.  \_4+V-49.  13.    V-12  +  4V-3. 

11.  V^^+V^=^.  14.   5V^^nj->/^r72. 


12.   2V-4  4-3V-1.  15.   .SV-20--V-80. 


270  IMAGINARY   NUMBERS 


16.  V  - 1 6  w^x-  4-  V  —  aV  -  V  —  9  a  V. 

17.  (V^^  +  3V"^)H-(V^^-3\/^^). 


18.    (V—  9  «?/  —  V  —  xy )  —  (V—  4  a??/  +  V—  xy). 


19.    V-o^^-f- V-4ar^-V-ar''4-3ajV- 


20.  V-16-3V-4  + V-18-f  V-50  + V-25. 

21.  V^S4-aV^^-V-98-5V^^a-. 


22.    V1-O-3V1-104-2V5-30. 


23.    Multiply  3  V  - 10  by  2  V  -  8. 

PROCESS 

3V^=l0  X  2  V^=^  =  3  VlO  V^=I  X  2  V8  V^^ 


-(W10x8x(-1) 

,       =-6V80=-24V5 

Explanation.  —  To  determine  the  sign  of  the  product,  each  imaginary 
number  is  reduced  to  the  form  6  V—  1.  The  numbers  are  then  multiplied 
as  ordinary  radicals,  subject  to  (A),  §  369.  that  V—  IxV—  1=  —  1. 

24.  Multiply  V^^  +  3  V^^  by  4  V^=^  -  V  -^. 

First  Solution  Second  Solution 

v'^^^  +  SV^^  =  (  \/2  +  3\/3)  V^=~T,  V^^  +  3  V^=^ 

4^"^=^-  v'^=^=(4V2  -  \/3)\/^=T;  4\A=^- V^^ 

...  (V^^+3\/^^)(4\/^=^-V^^^)  -4V4-12v^6 

=  {V2  +  3\/3)(4\/2  -  \/8)(\/^)2  +8V9+       V6 

=  (8  +  12V6-  \/6-9)(- 1)  1       -  IIV'6 
=    1  -llV6. 

Multiply : 

25.  3 V"^=^  by  2V^5.  28.    8 V^oC  by  V^=^. 


26.    4V-27bvV-12.  29.    V- 125  by  V- 108. 


27.    2V-8by5V-3.  30.    V-lOOby  V-30. 

31.    V^^+  V^^  by  V^=lJ  -  V^^. 


IMAGINAUY   NUMBERS  271 

32.    \  —  a^  +  V  —  a  by  \/  —  ab—  \/  —  a. 


33.    \  —xy-^  y/—xby  V  —  ay  +  V  —  x. 


34.  V -50- N/-12by  V-8-V-76. 

35.  \— <»-f-\       />-}-V  —  cby  V  —  a-fV— ^  —  V—  c. 

36.  Divide  v  -  12  by  V^. 


SOLDTION 

37.    Divide  V12  by  V^. 

Solution 


Vi  =  2. 


V-  3     VSV-  1     V^     V- 1 


-  1 
38.    Divide  5  by  (V— 1)*. 

Solution 


5 


=  -^^±M-  =  ^-i2^^Jy=6y/~l, 


(>/^^)*     (V-l)«  (V-T)" 
Divide : 

39.  V^TS  by  V^^.  46.    -  2  by  V=T. 

40.  V27  by  V^^.  47.    (V^=l)*  by  ^V^T. 

41.  14  v'^  by  2V^^.  48.    (V^)*  by  (V^n!)". 

42.  -  V^a«  by  V^^^.  49.    V4a6  by  V^^^. 

43.  1  by  ^/^^.  60.    (VITl)"  by  -^V^^T. 

44.  VS-f  3VT4by  V^^.  61.    (V^^"  by  ( V^rT)-«. 

45.  Vi2+V3byV^^.  52.    \/^^+6\/^by  V^^^. 

63.    V^^by  V^^- V^^- V^^. 


'272  KE  VIEW- 

REVIEW 

372.  1.  Distinguish  between  an  equation  and  an  identity  ; 
between  an  integral  and  a  fractional  equation.     Illustrate. 

2.  When  is  a  literal  equation  an  identity  ? 

3.  State  what  is  meant  by  a  graph.     Of  what  practical  use 
are  graphs  ? 

4.  Define  abscissa;  ordinate;    coordinates.     Interpret  the 
equation  ^  =  (—  4,  3). 

5.  Tell  how  to  determine  where  a  graph  crosses  the  ic-axis ; 
the  y-sixis. 

6.  Construct  the  graph  of  2y  ==  3  x  —  4. 

7.  Why  are  simple  equations  sometimes  called  linear  equa- 
tions ? 

8.  State    the  law    of    signs    for  involution;  the    law    of 
exponents. 

9.  How  may  the  involution  of  a  trinomial  be  performed  by 

the  use  of  the  binomial  theorem  ? 

10.  Tell  how  the  12th  root  of  an  expression  may  be  obtained. 

11.  Define  root  of  an  equation  ;  equivalent  equations ;  si- 
multaneous equations ;  independent  equations ;  indeterminate 
equations ;  elimination  of  an  unknown  number. 

12.  Upon  what  axiom  is  elimination  by  addition  based  ? 
elimination  by  comparison  ? 

13.  Define  radical ;  radicand  ;  surd ;  conjugate  surds.  Illus- 
trate each.  Is  ^2-f  Vi  a  surd?  State  reasons  for  your 
answer. 

14.  What  are  similar  radicals  ?     Illustrate. 

15.  Tell  what  is  meant  by  the  principal  root  of  a  number. 
What  is  the  principal  square  root  of  4?  the  principal  cube 
root  of  -8? 


REVIEW  278 

16.  Represent  VfO  inches  exactly  by  a  line. 

17.  Show  the  difference  in  meaning  between  (a*)*  and  a*  x  of. 

18.  What  does  the  numerator  of  a  fractional  exponent  indi- 
<;ite?  the  denominator ? 

19.  Show  that  a^  =  1 ;  that  — ,  =  aV. 

20.  Factor  a~*-|-2a"*6~* +  6'*,  giving  reasons  for  each  step. 

21.  Define  and  illustrate  mixed  surd ;  entire  surd.  Tell  how 
to  reduce  a  mixed  surd  to  an  entire  surd. 

22.  What  is  a  radical  equation  ?  Give  the  steps  in  the 
solution  of  such  an  equation. 

23.  Define  real  number;  imaginary  number ;  rational  num- 
ber; irrational  number. 

24.  ('lassify  the  following  numbers  as  real  or  imaginary  ; 
I  ^  rational  or  irrational : 

2,  V4,  V2,  -/5,  V^2,  V^  V^«,  v'^    <r^,   ^ 
"  being  a  positive  number. 

25.  Illustrate  how,  in  finding  the  value  of  an  expression 
with  an  irrational  denominator,  it  is  advantageous  to  ration- 

[  alize  the  denominator  first. 

26.  Find  the  value  of  i*,  i^,  /®,  i*.  What  is  the  value  of  even 
powers  of  i  ?  of  odd  powers  of  i  ? 

27.  Solve  graphically  the  simultaneous  equations 
2x-3.v=10, 
5x-f  2y  =  6. 

28.  If  a  system  of  two  linear  equations  is  indeterminate. 
how  will  the  fact  be  shown  by  the  graphs  of  the  equations, 
referred  to  the  same  axes  ?  how,  if  they  are  inconsistent  ? 

29.  When  a  is  positive,  is  V  —  a  real  or  imaginary  ?  When 
a  is  negative,  is  V— a  real  or  imaginary? 

30.  Find  the  sum  and  the  product  of  2V— 4  and  SV—  9. 


31.    Subtract  aA^  from  V-81;  divide  V-81  by  V-9. 
milne'r  stand,  alo.  —  18 


274  REVIEW 

EXERCISES 

373.    Reduce  to  simplest  form  : 

6o(^  —  7x^  —  5x  X  —  y     y +  ^      4a^2/2 

X  -\-  y     y  —  X     x^  —  y* 

V2-V3      V2  +  V3 


9x^-25. 

» 

8a;^+18.i-- 

-5 

Ux'-^r^x- 

-2 

cL^x^  —  a  ^x  -\-  x 

Vx 

a'-2aVb- 

\-h 

a-^h 

V2-V3- 

V5 

V2  4-  V3  + 

V5 

2  -  V5      2 

;Va 

2.     _.    .       .  .. 

V2  + V3      V2- V3 

4      -       ■   -  1  1   ,  1 


Va+V6  Vo  — V6 


6.    ^^-^ ~ L^.  11.     Vl+o^-Vl-.^ 


Vl4-a.-'^+ Vl 


.r 


6.    ^^ ^4-^^-^^-         12.    ^  + \=-^      ^ 


2+V5       V243  ■    1-V2a;      1  +  V2a;      1-2.1- 


,  o     ^^  +  V  a^  —  a^     07  —  Va^  —  a^ 

X  —  ^x^  —  d^      x-\-  -y/x^  —  d^ 


.     14     V^  H- 1  +  Va  —  1  _  V  g  +  1  —  Vet  —  1 
Va  +  1 -  Vtt-1      Va  +  1  +  Va— 1 

*       15  a^-6  a2_4aV5-|-46 

a^  -  2  a  V^  +  6      a-  +  2  a  v  ?>  +  6 


l  +  a  +  a^  ^  a     .  v'a;\/  a        V 


17.     1±^^.  19 


1  —  yg 4- a  /_g V^V_g_  _  Va;' 

Va  Vv' ^       «  AV^       ' 


REVIEW  275 

Expaud : 

20.    (a"*-^y.  24.  (a-'+a-y.  30.  (a  -  Vft/. 

26.  (ai-6i)«.  32.  (V2-V3V- 

^^^'    (i~l)*  27.  (a*-6-i)^  33.  (V5-2)«. 

28.  (a-^_/r^)«.  34.  (^-\/2)». 

29.  (a*  +  6V-  36.  ( V2  -  \/2)«. 


23. 


(-3-- 


Extract  the  square  root  of : 

37.  •^-H4y*+^-2a^  +  |-y2;. 

38.  a^  H-  12  ah^  +  54  a6  +  108  a^6^  +  81  b\ 

39.  1 -f  2\/«  — «— 2»V;r-f-.c^. 

40.  (i-\-4b-\-9c  —  4: Vah  -\- 6  Vac  —12  V6c. 

41.  .r--4a;Vary  +  6iFy  — 4  2/V«2^4-/. 

Find  the  square  root  of : 

42.  81234169.  46.  5fi  +  14Vl^l 

43.  64064016.  47.  47-12^/l^>. 

44.  .00022801.  48.  62 -f 20x6. 

45.  .1  to  four  places.  49.  51-36V2. 

Extract  the  cube  root  of : 

50.  .r'-9a;+27a;-»-27x-«. 

51.  27ur*-h27a;2-5  +  -i- - 


3  a!*     27aj» 

52.    a^  +  3aj*>/i-5a;\/«  +  3Vx-l. 


276  REVIEW 

53.  Find  the  cube  root  of  2  V2  -  6  v'2  -|-  3  V2  -v^i  -  2. 

54.  Extract  the  cube  root  of  510,082,399. 

55.  Extract  the  cube  root  of  1,042,590,744. 

56.  Extract  the  cube  root  of  2  to  three  decimal  places. 


57.  Find  the  first  four  terms  of  Vl  +  a;  —  a^. 

58.  Find  the  first  three  terms  of  Vl  +  a^. 

59.  Find  the  fourth  root  of 

tt"  -  4  a'  VoF"  +  6  a'b-^  -  4  ab-WoF'^  +  b-'. 

60.  Find  the  sixth  root  of 

8  -  48  Va  4- 120  a  -  160  a  V'a  + 120  a-  -  48  a-  Va  +  8  a' 

If  a"'  X  (<"  =  «'""''*•  for  all  values  of  m  and  «,  show  that : 

.      1  64.    (aby=l. 

61.  a--  =  -.  ^     ^ 

a- 

62.  a^  =  V^'=(Vaf'.  ^5.    (abcf  =  a'b^c'. 


9 


3/— o 


a-  ^^     /a\'     a 


63.    2(1-^  =  ^^-^^-  66.    r^     = 


a  \bj       W 

Find  the  value  of : 

67.  16^  70.    (aV)'.  73.    (W%)'^- 

68.  273.  71.    (6y)-l  74.    (|6)-l. 

69.  8"i  72.    (a'^6'')    ".  75.    (— o^)"*. 

76.  For  what  values  of  n  is  («  —  ft)"  =  {b  —  a)**  ? 

Simplify,  expressing  results  with  positive  exponents : 

77.  (36  a-'= -^  25  f/--)-^.  80.    (VoV^-- -v^a^aP)^ 

78.  (8  aV  X  64  a-^.i-T^.  81.    (V^r^-^  Va26)-i 

7 9 .  (^«  -^  b^) ^  -J-  (a^ft  ^) I  82 .    ( Va  -;-  Va)  h-  Va. 


REVIEW 


277 


83. 


a  —  b 


a  4- 6    .    2aU 


J  _  5*      a*  -1-6*     a^  -  ft^ 


.-;j 


1  -h  a-^6  ^  /I  +  ab-'  -h  «-/>-^'      1-ha 
•    1  -  a-'b  '  [l  -  ixb-'  -H  arb-''      1  -  a-«6« 

Solve  the  following  equations  : 

x±l  _ x  —  1  __ ^--6x 
'  x-1    .1-  -h  i     l-x"' 


86. 


87. 


88. 


89. 


l-2x     2.r-l      a;^5g-6.2      17  +  33- 
5  2  2.r  30 


10 


4.r-17     3|-22a;^^     6/j  _u 


33 


('  -«> 


3  x  -  5  v  _  2  a;-8y-9  _y       7 
5  ~T2  ~2"^12' 


(|,|^h)_^(4._|_24)=0. 


3x-M=22/, 

(x  +  5)(y  +  7)  =  (./•-+  l)(v-9)-Hll2. 


Simplify,  expressing  results  with  positive  exponents: 


90. 


L>7  "^  8  y 


93. 


3a-»-|-2x 


r 

94.     i(aW)*-!- (a- '/;)--(*. 


91.    ja-*[a*(aV]*i*- 

fey 


a-^b 


m 


(ax-y 


96 


b-\-a 
ab 


278  REVIEW 

Exercises  on  this  page  are  from  recent  examination  papers. 

97.  Write  the  first  five  powers  of  V—  1. 

i  ^        JL 

98.  Prove  that  (a'")"  =  a'"";  that  a"*  x  a-"  =  a'"-". 

1 

99.  Which  is  the  greater,  2^  or  5  ?     Prove  it. 

100.  Find  the  value  of  r to  two  decimal  places. 

2  +  V5  -  V2 

101.  Extract  the  square  root  of 

x(x  -  V2)(x  -  VS)(x  -  Vl8)  +  4. 

102.  Find  a  factor  that  will  rationalize  x^  -{-y"^. 

Simplify : 

103.  16^  •  2^  .  32^  106.    ^4=^  •   \/S  •  3\/4. 


104.    \38-12VlO.  107.    V-27  +  (V-l)'  +  8"-^. 


105.    V^-V^--^.  108.      /V23  +  V7, 

Vx-\-^x-2  \V"23-V7 

109.    12^ +  4^-9-^+       ^       +27i 
V-64 


110. 


111. 


112. 


V3  +  V2  .    7  +  4  V3 
2-V3     '  V3-V2 


\3  +  V5  -  Af3  ^  V5 
\3+V5  4-\3-^/5 

2V15  +  8  .  8V3-6V5 
5  -h  Vlo    '  5V3-3V^ 


113.  Ax  .rZx-^v^ 


114. 


l  +  a;      a;^  +  3a;  +  2 
(■x-\-2)-'-.{x-\-l)-\2-\-x)-' 


QUADRATIC   EQUATIONS 


374.  The  equation  a;  —  2  =  0  is  of  the  first  degree  and  has 
one  root,  x  =  2.  Similarly,  a;  —  3  =  0  is  of  the  first  degree  and 
has  one  root,  a;  =  3.  Consequently,  the  product  of  these  two 
simple  equations,  which  is 

(iC-2)(a;-3)=0,  or  or^  -  5  j; -f- <>  =  0, 

is  of  the  second  degree  and  has  two  roots,  2  and  3. 

375.  An  equation  that,  when  simplified,  contains  the  square 
<»f  the  unknown  number,  but  no  higher  power,  is  called  an 
(•(juation  of  the  second  degree,  or  a  quadratic  equation. 

It  is  evident,  therefore,  that  quadratic  equations  may  be 
of  two  kinds  —  those  which  contain  only  the  second  power  of 
the  unknown  number,  and  those  which  contain  both  the  second 
and  first  powers. 

X-  =  I  •')  and  ax^  -{■  bx  =  r  are  quadratic  equations. 

PURE    QUADRATIC   EQUATIONS 

376.  An  equation  that  contains  only  the  second  power  of 
the  unknown  number  is  called  a  pure  quadratic. 

ax^  =  b  and  ax-  —  cjti^  =  be  are  pure  (luadraiics. 

Pure  quadratics  are  called  also  incomplete  quadratics,  because 
they  lack  the  first  power  of  the  unknown  number. 

377.  Since  pure  quadratics  contain  only  the  second  power  of 
the  unknown  number,  they  may  be  reduced  to  the  general  form 
(^jr^  =  6,  in  which  a  represents  the  coefficient  of  x*,  and  b  the 

um  of  the  terms  that  do  not  involve  a^. 

279 


280  QUADRATIC    EQUATIONS 

378.  The  equation  3xr=  300  has  two  roots,  for  it  may  be 
reduced  to  the  form  (x—  10)(a;  -}-]0)  =  (».  which  is  equivalent 
to  the  two  simple  equations, 

ic  _  10  =  0  and  x-  + 10  =  0, 

each  of  which  has  one  root. 

The  roots,  +  10  and  —  10,  are  numerically  equal  but  opposite 
in  sign. 

Principle.  —  Every  pure  quadratic  equation  has  two  roots, 
numerically  equal  but  opposite  in  sign. 

It  is  proved  in  §  436  that  every  quadratic  equation  has  two  roots  and 
only  two  roots. 

EXERCISES 

379.  1.    Given  10  x^  =  99  -  x^,  to  find  the  value  of  x. 

Solution 
10  a;2  =  99  -  x2. 
Transposing,  etc.,  11  ic^  =  99. 

Dividing  by  11,  a;^  =  9. 

Extracting  the  square  root  of  each  member,  §  289, 

ic  =  ±3. 

Note.  —  Strictly  speaking,  the  last  equation  should  be  i  a;  =  ±  3, 
which  stands  for  the  equations,  4-a;=+3,  4-x  =  — 3,  and  —  a:  =  —  3,  and 
—  X  =  +  3.  But  since  the  last  two  equations  may  be  derived  from  the 
first  two,  by  changing  signs,  the  first  two  express  all  the  values  of  x. 
For  convenience,  the  two  expressions,  x  =-\-  Z  and  x  =  —  3,  are  written 
x=±3. 

Consequently,  in  extracting  the  square  roots  of  the  members  of  an 
equation,  it  will  be  sufficient  to  write  the  double  sign  before  the  root  of 
one  member. 

2.    Find  the  roots  of  the  equation  3a^=  —  15. 

Solution 

3x2= -16. 
Dividing  by  3,  x^  =—  5. 

Extracting  the  square  root,        x=±  V—  5. 

Verification.  —  The  given  equation  becomes  —  15  =  —  15  and  is  there- 
fore satisfied  when  either  -|-  V^^  or  —  V—  5  is  substituted  for  x. 


QUADRATIC   EQUATIONS  281 

Solve  for  x,  and  verify  each  root : 

3.  .Sar*- 0  =  1^2.  9.  7'        j:.  =  6a^H-73. 

4.  2a?*  +  3ar»  =  80.  10.  (./ •+ I  r  =  8a;4-26. 

5.  4a-2=:f  11.  (a-^)-=(3.c4-a)(a;-a). 

12.  aa^=(a-6)(a=''-6*)-^- 

13.  a^x'-^2a3^  =  {a'-lY-x'. 

14.  (a;4-2)«-4(x-h2)=4. 

21.  «  +  ^  =  ^. 
«     a      x 

22.  ^^ ?^  =  0. 

a  +  b        X 

23.  *-2     ^  +  2        40 


6. 

|ar^-5=22. 

7. 

r*-6  =  0. 

8. 

6aaj»-64a*  =  0. 

15. 

x-8         6 
6         x  +  8 

16. 

1      ,      1      _8 

1-a;      1-har     3 

17. 

12         5a;         6 

18. 

a;  +  3     a;-3_4 

x-3'x+3 

.r4-2     2-x     a^-4 


24     V^^  +  l-V-^Jzl^l, 

V^qpr+Vx'-i     2 

19.    ^=^  +  ^±2  =  _i.  25.    ^±^  +  ^-^*-   2a 


aj  +  l      X— 1  X  —  a     xH-«      1  —  a 

x-3      x-f  3_-,-  x-l-g     x-a_a*  +  &' 

..•-2"^x  +  2~    *•  a:+6"^x-6      x^^-ft'' 


27.    V(x  +  3)(x-5)=V49-2x. 


28.    V25-6x+V26-|-6x=8. 

29       ^4-7  x-7    ^      7 

•    x«-7x     x*-f7aj     a?-73 


Vx  +  2a —  Vx  —  2  a      x 

30.    p==  =  cr~ 

Vx-2a  +  Vx+2a     2  a 


31.    H =.r. 

j._l_V2-x*      x-\/2-x* 


282  QUADRATIC   EQUATIONS 

Problems 
380.   1.    What  negative  number  is  equal  to  its  reciprocal  ? 

2.  If  25  is  added  to  the  square  of  a  certain  number,  the 
sum  is  equal  to  the  square  of  13.     What  is  the  number  ? 

3.  What  number  is  that  whose  square  is  equal  to  the  dif- 
ference of  the  squares  of  25  and  20  ? 

4.  If  a  certain  number  is  increased  by  5  and  also  decreased 
by  5,  the  product  of  these  results  will  be  75.  What  is  the 
number  ? 

5.  How  many  rods  of  fence  will  inclose  a  square  garden 
whose  area  is  21  acres  ? 

6.  The  area  of  one  side  of  a  drawing  board  is  5  square  feet. 
If  it  were  3  inches  shorter  and  3  inches  wider,  it  would  be 
square.     Find  its  length  and  width. 

7.  The  sum  of  two  numbers  is  10,  and  their  product  is  21. 
What  are  the  numbers  ? 

Suggestion.  —  Represent  the  numbers  by  5  +  x  and  5  —  x. 

8.  The  sum  of  two  numbers  is  16,  and  their  product  is  55. 
What  are  the  numbers  ? 

9.  The  sum  of  two  numbers  is  5,  and  their  product  is  —  14, 
W^hat  are  the  numbers  ? 

10.  Factor  a^  +  17  a  4-  60  by  the  method  suggested  in  the 
preceding  problems. 

Suggestion. —  We  need  the  two  factors  of  60  whose  sum  is  17.  Repre- 
sent them  by  \'  +  x  and  y  —  x.     Then,  (J^^  +  x)  {}^'  -  x)  =  60. 

11.  Separate  a^  -f  2  a  —  2  into  two  factors. 

12.  Separate  a^  —  2  ic  —  1  into  two  factors. 

13.  Divide  24  into  two  parts  whose  product  is  143. 

14.  The  sum  of  the  squares  of  two  numbers  is  394,  and  the 
difference  of  their  squares  is  h^.     What  are  the  numbers  ? 


QUADRATIC    EQUATIONS  288 

16.   The  length  of  a  10-acre  field  is  4  times  its  width.   What 

r»'  its  (limensious? 

16.  At  75  cents  per  square  yard,  enough  linoleum  was  pur- 
chased for  $30  to  cover  a  rectangular  tioor  whose  length 
was  3  times  its  breadth.  Whatvere  the  dimensions  of  the 
tioor? 

17.  The  lock  of  the  St.  Mary's  Canal,  Michigan,  is  8  times 
:i8  long  as  it  is  wide ;  the  surface  of  the  water  it  contains  is 
S(),000  square  feet  in  area.  What  are  the  dimensions  of  the 
lock? 

18.  A  man  has  two  square  fields  that  together  contain  51J 
acres.  If  the  side  of  one  is  as  much  longer  than  50  rods  as 
that  of  the  other  is  shorter  than  HO  rods,  what  are  the  dimen- 
sions of  each,  field? 

19.  A  man  had  a  rectangular  field  whose  width  was  J  of  its 
length.  He  built  a  fence  across  it  so  that  one  of  the  two  parts 
thus  formed  was  a  square.  If  the  square  field  contained 
10  acres,  what  were  the  dimensions  of  the  original  field  ? 

20.  A  shipment  of  railroad  ties  measuring  400,000  board 
ieet  contained  as  many  car  loads  as  there  were  board  feet  in 
a  tie.  If  each  car  held  250  ties,  find  the  total  number  of  ties 
and  the  number  of  board  feet  in  one  tie. 

Formulae 
381.    Solvj'  the  following  formula^  tiom  pliysics: 

1.  .  =  !,gtM.,vl.  ^    F="'f.  forv. 

2.  E  =  KMi^.  toTv.  '' 

6.  When  g  =  32.16,  formula  1  gives  the  number  of  feet  (s) 
through  which  a  body  will  fall  in  t  seconds,  starting  from 
rest.  How  long  will  it  take  a  brick  to  fall  to  the  sidewalk 
from  the  top  of  a  building  100.5  feet  biuli  ' 


284 


(QUADRATIC    EQUATIONS 


7.  To  lighten  a  balloon  at  the  height  of  2500  feet,  a  bag  of 
sand  was  let  fall.  Find  the  time,  to  the  nearest  tenth  of  a 
second,  required  for  it  to  reach  the  earth. 

Solve  the  following  geometrical  formulae  : 

8.  c2  =  a-  +  h\  for  h.  10.    A  =  .7854  cJ^,  for  d. 

9.  4  m-  =  2(a-  +  b-)  -  (r,  for  m.  11.    V  =  \  iri^h,  for  r. 

12.  Using  formula  8,  find  the  hypotenuse  (c)  of  a  right  tri- 
angle whose  other  two  sides  are  a  =  8  and  ^  =  6. 

13.  Solve  formula  8  for  b,  and  find  the  side  (6)  of  a  right 
triangle  whose  hypotenuse  (c)  is  5  and  whose  side  (a)  is  3. 

14.  From  formula  8  and  the  accompanying  figure  find,  to 

the  nearest  tenth,  the  side  (a)  of  a  square 
inscribed  in  a  circle  whose  diameter  (d) 
is  10. 

15.  Find,  to  the  nearest  tenth  of  an  inch, 
the  dimensions  of  the  largest  square  timber 
that  can  be  cut  from  a  log  12  feet  long  and 
18  inches  in  diameter. 

16.  Using  formula  8  and  the  accompany- 
ing figure,  deduce  a  formula  for  the  alti- 
tude (/i)  of  an  equilateral  triangle  in  terms 
of  its  side  (c). 

17.  From  formula  9,  find  the  length  of  the 
median  (m)  to  the  side  (c)  of  the  triangle  in 
the  accompanying  figure,  if  a  =  11,  Z>  =  8,  and 
c  =  9. 

18.  Substituting  in  formula  10,  find,  to  the  nearest  tenth  of 
a  foot,  the  diameter  (d)  of  a  circle  whose  area  {A)  is  1000 
square  feet. 

19.  Using  formula  11,  find,  to  the  nearest  centimeter,  the 
radius  (r)  of  the  base  of  a  conical  vessel  20  centimeters  high 
Qi  =  20)  that  will  hold  a  liter  of  water  (F=  1  liter  =  1000 
cu.  cm. ;    IT  =  3.1416). 


QUADRATIC   EQUATIONS  285 

AFFECTED  QUADRATIC  EQUATIONS 

382.  A  quadratic  e(iuatioii  that  contains  both  the  second  and 
I  lie  first  powers  of  one  unknown  number  is  called  an  affected 
quadratic. 

r-  +  ;ix  =  10,  4  X-  —  X  =  3,  and  ax-  +  ftx  +  c  =  0  are  afft-clcd  (iu;i<h-;Uics. 

Affected  quadratics  are  sometimes  called  complete  quadratics. 

383.  Since  affected  quadratic   equations   contain   both  the 
t'cond  and  the  first  powers  of  the  unknown  number,  they  may 

always  be  reduced  to  the  general  form  of  cijr  -\-  bx  -\-  c  =  0,  in 
which  a,  b,  and  r  may  represent  any  numbers  whatever,  and 
Xy  the  unknown  number. 

The  term  c  is  called  the  absolute  term. 

384.  To  solve  affected  quadratics  by  factoring. 

Reduce  the  equation  to  the  form  ax-  +  6x'  -f  c  =  0,  factor  the 
first  member,  and  equate  each  factor  to  zero,  as  in  §  172,  thus 
obtaining  two  simple  equations  together  equivalent  to  the  given 
(piadratic,  subject  to  the  exceptions  given  in  §  230  as  to 
equivalence. 

Thus,  3  x^  =  10  x  -  ;^. 

Transposing,  8  x*  -  10  x  +  3  =  0. 

Factoring,  (x  -  3)  (3  x  -  I )  =  0. 

.-.  x-3  =Oor:]x  -  1=0; 
whence,  r  =  3  or  i. 

EXERCISES 

385.  Solve  by  factoring,  and  verify  results : 

1.  .r  -  5  a-  4-  ^J  =  0.  7.  'J  x'  —  7  X  -\-  :>  =  0. 

2.  r'  4- 10  J-  +  21  =  0.  8.  2  z-  -  z  -  3  =  0. 

3.  a^  -h  12  J-  -  28  =  0.  9.  .'U**  -  2  V  -  8  =  0.  . 

4.  a^-20a- 4-'">l  =0.  lo.  10  r*  -  27  r  +  o  =  0. 

5.  a^  -  5  J-  =  24.  11.  fir.r  +  1)  =  l-"^  •'<• 


6.    a-  -  1  =  3(.r  -f-  1 ).  12.    2  .r  -f-  7  a;  =  4. 


286  QUADRATIC    EQUATIONS 

386.  First  method  of  completing  the  square. 
Since  (x  -}- ay  =  ocr  -^  2  ax -\-  a\ 

the  general  form  of  the  perfect  square  of  a  binomial  is 
x^  -\-2  ax  +  a^. 

Consequently,  an  expression  like  x--\-2ax  may  be  made  a 
perfect  square  by  adding  the  term  a-,  which  it  will  be  observed 
is  the  square  of  half  the  coefficient  of  x. 

Thus,  to  solve  y? -\-^x=  —  5 

by  the  method  of  extracting  the  square  root  of  both  members 
(the  method  used  in  solving  pure  quadratics),  we  must  com- 
plete the  square  in  the  first  member. 

The  number  to  be  added  is  the  square  of  half  the  coefficient 
of  X ;  that  is,  (|)^,  or  9.  The  same  number  must  be  added  to 
the  second  member  to  preserve  the  equality. 

Therefore,  Ax.  1,  .^•^  +  6  .t  +  9  =  -  5  4-  9  ; 

that  is,  .c2  -f  6  .-r  +  9  =  4. 

Extracting  the  square  root,  §  289,    .t  +  8  =  ±  2  ; 
whence,  x—  —  3  +  2  or  —  3  —  2. 

.-.  x=  —1  or  —  o. 

EXERCISES 

387.  1.    Solve  the  equation  yr  —  5  it*  —  14  =  0. 

SOUITIOX 

ir2  -  .-)x -14  =0. 
Transposing,  .r^  —  5  x  =  14. 

Completing  tlie  square.         x?'  —  bx  Ar  '-^~  ==  14  +  '^-^  =  ^-. 
Extracting  the  square  root,  x—  I  —  ±  ^; 

whence,  ,c  =  f  +  |  or  |  —  |. 

.-.  ./•  =  7  or  —  2. 

Verification.  —  Either  7  or  —  2  substituted  for  x  in  the  given  equa- 
tion reduces  it  to  0  =  0.  an  identity  ;  that  is.  the  given  equation  is  satis- 
fied by  tliese  vahies  of  x. 


.     QUADRATIC    EQUATIONS  287 

2.    Solve  the  equation  4  jr  -f  4  x  -f-  6  =  0. 

Solution 
4  a;^  +  4  X  +  6  =  0. 
Tramsposing,  4  x^  +  4  ac  =  —  6. 

Dividing  by  4,  x^  +  x  =  —  |. 

Completing  the  square,       x'^  +  x  +  ^=-§  +  ^  =  —  f. 
Kxtracting  the  square  root,        J'+i  =  ±jV—  6- 

.\x-=  -J  +  iV^or  -  .^-^V^), 
which  would  usually  be  written,  x  =  \{—  I  ±  V—  6), 

Steps  in  the  solution  of  an  affected  quadratic  equation  by 
the  tirst  method  of  completing  the  square  are : 

1.  Transpose  so  that  the  terms  containing  oj*  and  x  are  in  one 
member  and  the  known  terms  in  the  other. 

2.  Make  the  coefficient  of  a^  positive  uniti/  by  dividing  both 
members  by  the  coefficient  of  s?. 

3.  Complete  the  square  by  adding  to  each  member  the  square 
of  half  the  coefficient  ofx. 

4.  Extract  the  square  root  of  both  members. 

5.  Solve  the  two  simple  equations  thus  obtained. 

Solve,  and  verify  all  results : 

3.  2?-2x=U?K  12.  y2^10-3?/. 

4.  .r«-f-2a-  =  U>8.  13.  2--180  =  .S;2. 

5.  .f='-4u'=ll7.  14.  v2^15i;  =  54. 

6.  ar='-6a-=160.  15.  t--f21r  =  -54. 

7.  8x  =  ae'-180.  16.  n(n-l)  =  9.S(). 

8.  ar'-f-2a-  =  120.  17.  r^^  27  r  +  140  =  0. 

9.  a^ -1-22  J- =  -120.  18.  /--lU  +  28  =  0. 

10.  .jr=2H.r-  187.  19.    5x*-3x-2  =  0. 

11.  .r-'-12j;  =  189.  20.    6x*-5a;-6  =  0. 


'2SS  QUADRATIC    EQUATIONS 

21.  .2  it-' -h  .9  a;  = .')  o. 

22.  .03a;^-.07a-  =  .I. 

23.  2  ar  —  J^9- ;t- =  |. 

388.  Other  methods  of  completing  the  square. 

To  apply  the  first  method  of  completing  the  square,  the  coef- 
ficient of  a^  must  be  +1  or  be  made  + 1. 

Other  methods  of  completing  the  square  are  based  on  mak- 
ing the  coefficient  of  a?^  a  perfect  square,  if  it  is  not  already  one, 
by  multiplying  or  dividing  both  members  of  the  equation  by 
some  number. 

Thus,  given  3  x-  -|- 10  a;  =  -  3. 

Multiplying  by  3,      9  a^  +  30  x  =  -  9. 

When  the  square  of  the  first  member  is  completed,  30  a; 
will  be  twice  the  product  of  the  square  roots  of  the  terms  that 
are  squares.  Hence,  the  square  root  of  the  term  to  be  added  is 
15  a;  H-  V9  a^,  or  5 ;  and  the  number  to  be  added  is  5^,  or  25. 

Completing  the  square, 

9  ar^  +  30  a.'  +  25  =  -  9  -f  25  =  16. 

When  the  coefficient  of  x^  has  been  made  a  perfect  square,  the 
number  to  be  added  to  complete  the  trinomial  square  is  obtained 
by  dividing  half  the  term  containing  x  by  the  square  root  of  the 
term  containing  x^,  aiid  squaring  the  quotient. 

EXERCISES 

389.  1.    Solve  the  equation  8  x^  -  10  a;  =  3. 


SOLFTION 

8  a:2  -  10  X  =  3. 

Multiplying  by  2, 

16x2  -  20x  =  6. 

Completing  the  square, 

16  x2  -  20  X  +  V  =  fi  +  ¥-  =  ¥• 

Extracting  the  square  root, 

4X-|=:±|. 

4  X  r^  1  ±  1  =  6  or 

. •.  X  =  1  or  -I. 

QUADRATIC     EQUATIONS  289 

Solve,  aiid  verify : 

2.  2««-5.r  =  42.  7.  3j^  +  4x  =  95. 

3.  6x«-r,x  +  l=0.  8.  7v2-f2y  =  32. 

4.  4x*-12a;  =  27.  9.  8a^-lHa;  =  5. 
6.    lSx^  +  6x  =  4.  10.  6 7/1=' -I- 5/// =4. 

6.   2a^-U^H-12  =  0.  11.   5n--\i  H  =  ->^. 

12.   Solve  the  general  quadratic  equation  ax^  -h  to  +  c  =  0. 

Solution 

oac*  +  6x  +  c  =  0.  (1) 

Transposing  c,  ax^  +  bx=  —  c.  (2) 

Multiplying  by  a,  a'^'*  +  afca:  =  —ac.  (3) 

Completing  the  square,       aV^  +  afcx  -i —  = ac.  (4) 

4        4 

Multiplying  by  1 .  1  a-a--  +  4  a6a;  +  6*  =  6^  _  4  ar.  (6) 


Extracting  the  stiuare  root,  2ax  +  b  =  ±\  b-  — iac.  (6) 


...x  =  -*±^^^HP«.         (T) 
2a 

J         It  isevidfent  that  (5)  can  l)e  obtained  by  multiplying  (2)  by  4  «  and  add- 
[    ing  b^  to  both  members.     Hence,  when  a  quadratic  has  the  general  form 
'     of  (1),  if  the  absolute  term  is  transposed  to  the  second  member,  as  in  (2), 
the  square  may  be  completed  and  fractions  avoided  by 

MuUiplyiiKj  hy  4  timeH  the  coefficient  of  x^  and  adding  to  each  member 
fhf'  square  of  the  coefficient  of  x  in  the  given  equation. 

This  is  called  the  Hindoo  method  of  completing  the  square. 

Solve  by  the  Hindoo  method,  and  verify  results : 

13.  2a;*4-3a-  =  27.  18.    4  .i--a-- 3  =  0. 

14.  2ir*-|-6aj  =  7.  19.    r>  j^  -  2  j- -  IT)  =  0. 

15.  2jr  +  7j;=-r,.  20.    3^^4-7.1-110  =  0. 

16.  3j^-7.r=-2.  21.   2ar'-5a;-150  =  0. 

17.  4x-2-17x  =  -4.  22.    3r2-h.c-200  =  0. 

MILNE's   8TAX1>.    alo. —  1!» 


290  QUADRATIC    EQUATIONS 

23.    5a;2-7.r  =  -2.  25.    15ar^- 7  .'k-2  =  0. 

24:.    6x^-{-5x  =  -l.  26.    7  3^^-20  X- 32  =  0. 

390.    To  solve  quadratics  by  a  formula. 
The  general  quadratic 

ax^  -i-bx-\-c  =  0  (1 ) 

has  been  solved  in  exercise  12,  §  389.     Its  roots  are 


6  ±  V62  -  4  oc 


Since  (1)  represents  any  quadratic  equation,  the  student  is 
now  prepared  to  solve  any  quadratic  equation  whatever,  that 
contains  one  unknown  number. 

The  roots  of  any  quadratic  equation,  then,  may  be  obtained  by 
reducing  it  to  the  general  form  and  employing  (2)  as  a  formula. 

EXERCISES 

391 .    1.  Solve  the  equation  6  a^  =  a;  + 15. 
Solution.  — Writing  the  equation  in  the  general  form 
6x^-x-  15  =0, 
we  find  that   a  =  6,    b  =—  I,    and   c  =—  15. 


...  by  (2),  §  390,  X  =  l±V(^ir^-^x^(-l^) 
•^  ^  ^  ^        '  2x6 

..L±l?  =  §or-l 
12  .3  2 

Solve  by  the  above  formula,  and  verify  results : 

2.  2ar^+5.«  +  2  =  0.  10.  l-3a;  =  2ar^. 

3.  Sx^-{-llx-h6  =  0.  11.  4  =  a;(3a;  +  2). 

4.  6xr-^2  =  7x.  12.  x^-5x==-3. 

5.  4a^+4i«  =  15.  13.  3x'-6x  =  -2. 

6.  2x''  =  9-3x.  14.  4ar^-3a;-2  =  0. 

7.  x(2x-^3)=-l.  15.  a^  +  10  =  6x. 

8.  13a;  =  3.r2-10.       "  16.  x'^ -4:(x-{-S). 

9.  7ic2+9a;  =  10.  17.  4.(2x-5)  =  x'. 


QUADRATIC   EQUATIONS  291 

392.    Miscellaneous  equations  to  be  solved  by  any  method. 

EXERCISES 

1.  Solve  the  equation  3  ar*  -f  2  .r  =  0. 

Remark.  —  Dividing  by  x  removes  the  root  a;  =  0  and  reduces  the 
"qnation  to  the  simple  equation  3  a;  +  2  =  0,  whose  root  is  a;  =  —  \. 

If  the  given  equation  is  solved  by  quadratic  methods,  the  roots  are 
ioiind  to  be  the  same,  namely,  0  and—  \  ;  consequently,  it  is  important 
to  account  for  roots  that  may  be  removed  (§  230)  by  dividing  by  an  ex- 
pression that  involves  the  unknown  number.  The  root  removed  is  the 
root  of  the  equation  formed  by  equating  the  divisor  to  0. 

2.  Solve  the  equation  -^  -  ^±1^  =  4. 

X—  I  x^ 

Solution.  d£L_?i+A?  =  4. 

r-1  x* 

First  reducing  the  second  fraction  to  its  lowest  terms,  then  multiplying 
l30th  members  by  the  L.C.  D.,  x  (x  —  1),  simplifying,  etc.,  we  have 
jc2_2a:-3=0. 

Factoring,  rx  -  3 Vx  -t-  1 )  =  0. 

.  a;  =  3or  -  1. 

Vkrification.  —  When  x  =  3,  each  member  =  4  ;  when  x  =  —  1,  each 
member  =  4  ;  that  is,  both  x  =  3  and  x  =  —  1  are  found  to  be  roots  of  the 
given  equation. 

XoTE.  —  If  the  second  fraction  is  not  reduced  to  lowest  terms  before 
clearing  the  equation  of  fractions,  the  multiplier  is  x'^  (x  —  1 )  instead  of 
'■  'x—  1),  and  the  root  x=  0  so  introduced  must  be  rejected. 

In  general,  no  root  is  introduced  by  clearing  an  equation  of  fractions, 
,'i'nvided  that :  fractions  having  a  common  denominator  are  combined  ; 
■  'irh  fraction  is  expressed  in  its  lowest  terms ;  and  both  members  are  then 
uinltipUed  by  the  lowest  common  denominator. 

General  Directions. — 1.  Reduce  the  equation  to  the  general 
form  ax"-  -\-  hx  -|-  c  =  0. 

2.  If  the  factors  are  readily  seen,  solve  by  factoring. 

3.  If  the  factoi's  are  not  readily  seen,  solve  by  comjtl^fi^'o  f^"' 
.square  or  by  formula. 

4.  Verify  all  results,  reject  roots  introduced  in  the  proceati  of 
ipducing  the  equation  to  the  general  form,  and  account  for  roofs 
that  have  been  removed. 


292  QUADRATIC   P:QUATI0NS 

Solve  according  to  the  general  directions  just  given : 


4.    2  2if-5x  =  (). 


6.  a^  =  3ic4-10. 

7.  ir2-30  =  13a;. 


11.  4aj2-12u-  =  0. 

12.  072-4.3^^=27.3. 


15. 

16. 

X            x-2 

9(x-l)         6 

17. 

4              1 

a,2_2a;  +  l      4 

18. 

4        3 

2ar^4-a.'      0^-3 


--     l-{-x      x—1      4 


21        a^  ^~^_^, 

a;  — 5         a;         2 


8.  o--r2x  =  2S.  22.    ^±I+i^±i^  =  7. 

a;  +  5       07  +  6 

9.  a72_i2a;  =  0. 

oo     a;  +  4  (a;  +  3)\ 

10.    18x2  +  6.r  =  0.  ^^-    a;32"^^-  a^_9 


24.      -^     =     ^     +5. 

x-2     x-2^ 


13.  .*'^  +  .2oaj=.15.  ^5-    ^72'^2~    2:^-   ' 

14.  :.4-l_|  =  ().  26.    -^  +  ^±^  =  3. 

aj2  i»  +  7a;  +  3 

27.    ^±?-^^±-^  =  l. 
a;  —  7      a:  —  5 

28    ^-3  I  .T  +  2^23. 
•    a;  +  4      07-2      10 

2CC  +  1      5_x--8 


29. 


l-2ic      7         2 


30.    1^  =  2-       ^ 


af  —  3x  xr  —  Sx 

Find  roots  to  the  nearest  thousandth : 

31.  a,2_4^^_|^o.  33.   ,^2_|_5^^_^5.5^0. 

32.  v'  +  6v^7  =  0.  34.    ^--12^4-16.5  =  0.    - 


QUADKAl'lC    EQUATIONS  298 

Literal  Equations 

393.  The  methods  of  sohition  for  literal  quadratic  equations 
;iie  the  same  as  for  numerical  quadi*atics.  The  method  by 
factoring  (§  384)  is  recommended  when  the  factors  can  be 
seen  readily.  If  it  is  necessary  to  complete  the  square,  the 
tirst  method  (§  386)  is  usually  more  advantageous,  provided  the 

'efficient  of  a;*  is  +1,  otherwise  the  Hindoo  method  (§  389) 
>  better,  because  by  its  use  fractions  are  avoided.     Results 
may  be  tested  by  substituting  simple  numerical  values  for  the 
literal  known  numbers. 

EXERCISES 

394.  Solve  for  x  by  the  method  best  adapted : 

1.  7?  —  (tx  —  ah  —  hx.  4.    hx —  2 ax  =  x^ —  \0a. 

2.  x^ -\- ax  =  ojc -\- ex.  5.    .i^ -\- 'Mjx  = 'm-x -\- W  he. 

3.  j^=  {m  —  u)x-\-  mil.  6.    (\:r^ -\-^ax —  '2  hx -{-ah. 

7.  ncQi^  —  hex  —  hd-\-  adx  =  0. 

8.  .1^  +  4 mx  -\-  3  ux-\-\2  mu  =  0. 

9.   a^  =  4ax-l.'o'.  j^    ,.  +  "" ^ <<:  +  j. 

X       b 

10.  jr  —  ax  -h  a-  =  0. 

11.  4a.«-.*-  =  3o*.  18.    2x-' —  =  a-2x. 

a 

12.  5«.r  +  r,(r'  =  6a:*.  ^^         1      ^^      aa;-4 

13.  21  h'--ih.r  =  x'.  "•^•  +  '*  1^ 

20.    ^  +  ?x  =  «-t-^ 


14. 

7  m* 
12 

3 

15. 

36  = 

5a;     6 
4  "^3* 

16. 

X 

X 

ni. 

x-1 

.^4-1 

21.   ^  +  2  =  (2aLM).. 


22.   .^■,2x^4(a6-lX 
ah  ab 


294  QUADRATIC    EQUATlOiVS 

23.  X-  —  2(a  —  b)x  —  4:  ah. 

24.  ar^  —  2  x{m  —  n)  =  2  m?^. 

25.  x^  -I-  2(a  -f-  8)x  =  -  32  a. 

26.  ^2  _^  ,^  _|.  5^  ^  5  ^  ^(^a,  _|_  ly 

27.  a(2  a;  -  1)  H-  2  feic  -  6  =  a-(2  a;  -  1). 

28.  XT  -f  4(a  —  l)a;  =  8  a  ~  4  a^. 

a  -\-  0  -{-  X      a      0      X 
30.       ^''  +  1  1       -* 


ri'-a;  —  2  ?<-      2  —  wa;      n 

2a  +  a;  g  —  2a;^8 

'    2g-a.-  g  +  2a;      3* 

1  1  3  +  «2 


32. 


g  —  a;      g  H-  a;      g-  —  x' 

33.  g(a;  —  2  a  +  6)  +  g(>c  +  a  -  b)  =  x^  -  (a  —  bf. 

34.  -^-fl+J-V4-i  +  ^  =  0. 
g4-&      V         abj        a      b 


35. 

XT  -\-l      a-\-b         c 

X              c          a  -{-  b 

36. 

2  X  —  a      o         4  g 
b           "      2x-b 

37. 

bx        ^^_a(x-^2b) 
a  —x      '          g  4-  6 

38. 

3a;-f6           b        1          1 

X  -^b       2x  —  a      1   1  ^  ~"  ^ 

a-\-b 


39.  ■^+f..+£^Y-f-^+^y=«x. 

a^      \         x)       \a       xj 


QUADRATIC   EQUATIONS  295 

Radical  Equations 

395.  In  §§  364,  365,  the  student  learned  how  to  free  radical 
equations  of  radicals,  the  cases  treated  there  being  such  as  lead 
to  simple  equations.  The  radical  equations  in  this  cliapter 
lead  to  quadratic  equations,  but  the  methods  of  freeing  them 
of  radicals  are  the  same  as  in  the  cases  already  discussed. 

396.  The  principles  of  §  230  in  regard  to  the  equivalence  of 
•  ([nations  have  been  illustrated  in  §  392,  exercise  1,  showing 
the  removal  of  a  root  by  dividing  by  an  unknown  expression, 
and  exercise  2,  showing  the  introduction  oi  a  root  by  clearing  of 
tractions  unless  certain  precautions  are  taken.  In  the  dis- 
cussion of  §  366  it  was  shown  that  the  processes  of  rationali- 
-ntion   and   involution,  used   in    freeing   radical   equations   of 

adicals,  are  likely  to  introduce  roots  according  to  the   con- 
\  (nition  adopted  there  as  to  how  roots  shall  be  verified. 

Heace,  it  is  im})ortant  in  the  solution  of  equations  that  roots 
be  tested  not  only  to  determine  the  accuracy  of  the  work,  but 
to  discover  whether  the  solutions  obtained  are  really  roots  of 
the  given  equation,  and  also  to  examine  the  processes  era- 
l)loyed  in  reducing  equations  to  see  whether  any  roots  have 
been  removed. 

BXBRCI8B8 

397.  1.    Solve  the  equation  2Vx  —  x  =  x  —  8  Va*. 

Solution 
2\/3r-  x  =  x  —  Sy/x. 
Dividing  by  \^z,  2  —  Vx  =y/x  —  S. 

Transposing,  etc.,  y/x  =  6. 

Squaring.  x  =  2o. 

Verification.  —  When  x  =  26, 

Ibt  member  =  2  V25  -  26  =  10  -  2-'.  =       16  ; 
2d  member  =  26  -  8vz6  =  26  —  40  =  -  15. 

Hence,  x  =  2')  is  a  root  of  the  equation  ;  ar  =  0,  the  root  of  the  equation 
\  X  =0,  also  is  a  root  of  the  given  equation,  removed  by  dividing  both 
members  bv  Vx. 


296  QUADRATIC    EQUATIONS 


2.    Solve  and  verify  Va?  +  1  +  Va;  —  2  -  V2  a;  —  5  =  0. 
Solution 


Vx+  1+  Vx-2  -  a/2  a;  -  5  =  0. 


Transposing,  Vx  +  1  +  Vx  -2=  V2x  —  b. 


Squaring,        x  +  1  +  2  Vx^  -x  —  2+  x-2  =  2x-5. 


Simplifying,  Vx^  _  x  -  2  =  —  2. 

Squaring,  x^  —  x  —  2  =  4. 

Solving,  x=  —  2  or  3. 

Verification.  —  Substituting  —  2  for  x  in  the  given  equation, 

that  is,  V^^l  +  2  V^^  -  3  V  -~r=  0. 

Therefore,  —  2  is  a  root  of  the  given  equation. 
Substituting  3  f^r  x  in  the  given  equation, 

Vi  +  Vf  -  Vl  =  0, 

which  is  not  true  according  to  the  convention  adopted  in  the  discussion  in 
§  366.     Hence,  3  is  not  to  be  regarded  as  a  root  of  the  given  equation. 

Note.  — The  equation  could  be  verified  for  x  =  3  if  the  negative  square 
root  of  1  were  taken  in  the  second  term  and  the  positive  square  root  in 
the  third,  thus : 

V4+  VT- Vl  =  2  + (-!)-(+ 1)=0. 

This  is  an  improper  method  of  verification,  however,  for  it  has  been 
agreed  previously  that  the  square  root  sign  shall  denote  only  the  positive 
square  root. 

Solve  and  verify,  rejecting  roots  that  do  not  satisfy  the 
given  equation,  and  accounting  for  roots  that  otherwise  might 
be  lost : 

3.    8V^-8if  =  |.  5.    x-l-{--Vx  +  5  =  0. 


4.    3  oj  +  Va- =  5  V4a;.  6.    a?  — 5  -  Vaj  — 3  =  0. 

7.    V'4  a;  +  17  +  Va;  4- 1  —  -1:  =  0. 


8.    l  +  V(3-r).r)--H  16  =  2(3-37). 


QUADRATIC    EQUATIONS 


9.    \l-|-x-V^+12  =  l-|-u:. 


10.    Va--H- V2a;-1- V5a;  =  0. 


11.    V2a;  — 7  — V2ic+ Va;-7  =  0. 


12.  Va;  +  3+ V4a;  +  1- Vl0»  +  4=0. 

13.  Va-l-x  —  Va  —  x=  V2 x. 


14.  Vac  —  a  -h  V6  —  x  =  V^  —  a. 

16.  Va^-6*=  Vaj  +  6Va  +  6. 

16.  -^2^4- Vl0"«  +  i=  V2~ac4-1. 

17.  VH-j-ic  +  V«  —  VlO  —  4  «  =  0. 


18.    V4a?-3- V2a;-|-2=:  Va;-6. 


19.   V2X-4-3-  \/a;-|-l  =  VS" 


14. 


20.    V3  a:  —  5  +  Va;  —  9  =  V4  ic  —  4. 


21.    VaM-8 


6 


J". 


22.  a-H- Var  +  m*  = 

23.  x  +  V«*  -  a*  =  - 


2m2 


,2 


a' 


24.    2x+V4^-l^^ 
2a!- V4a^-i 


86.   J^^+»,^±«  =  a'. 
^a;  +  a      >a;  — a 


26. 


'o-Ta; 4-  Vft  —  a;  =  Va  +  6  —  2 a?. 


27.  Vx-l-a*-  Va;-2a*=  V2«-5a*. 

28.  Vwiw  —  ar  —  Va;  Vwiw  —  1  =  \  ///  //  \  1 


297 


298  Ql'AURATIC    EQUATIONS 

Problems 

398.    1.    The  sum  of  two  numbers  is  8,  and  their  product  is 

l.").      Find  the  numbers. 

Soli  TioN 

Let  ic  =  one  number. 

Then,  S  -  x  =  the  other. 

Since  their  product  is  16,         (8  —  x)x  =  15. 
Solving,  X  =  S  or  .'>, 

and  8  —  X  =  5  or  3. 

Therefore,  the  numbers  are  3  and  5. 

2.  Divide  20  into  two  parts  whose  product  is  96. 

3.  Divide  14  into  two  parts  whose  product  is  45. 

4.  Find  two  consecutive  integers  the  sum  of  whose  squares 
is  61. 

6.  A  rectangular  garden  is  12  rods  longer  than  it  is  wide 
and  it  contains  1  acre.     What  are  its  dimensions  ? 

6.  A  plumber  received  $24  for  some  work.  The  number  of 
hours  that  he  worked  was  20  less  than  the  number  of  cents  per 
hour  that  he  earned.     Find  his  hourly  wage. 

7.  The  1860  bunches  of  asparagus  from  an  acre  of  land 
were  sold  in  boxes  each  holding  1  less  than  ^  as  many  bunches 
as  there  were  boxes.     Find  the  number  of  bunches  in  a  box. 

8.  Some  boys  laid  out  basket-ball  grounds  30  feet  greater 
in  length  than  in  width,  but  in  order  to  bring  the  area  down  to 
the  prescribed  limit  of  3500  square  feet,  they  reduced  the 
length  10  feet.  How  much  too  large  had  they  laid  out  the 
grounds  ? 

9.  The  volume  of  a  standard  size  of  concrete  building 
blocks  was  2880  cubic  inches.  The  smallest  dimension  was 
9  inches,  and  the  greatest  was  13  inches  more  than  the  sum  of 
the  other  two.     What  were  the  three  dimensions  ? 


C^LADRATKJ    KQU AXIOMS  2i^y 

10.  A  pai-ty  hired  a  coach  for  $12.  In  consequence  of  the 
tailure  of  three  of  them  to  pay,  each  of  the  others  had  to  pay 
20  cents  more.     How  many  persons  were  in  the  party  ? 

Solution 

Let  X  =  the  number  of  persons. 

Theii.  X  —  3  =  the  number  that  paid, 

—  =  the  number  of  dollars  each  should  have  paid, 

X 

'  =  the  number  of  dollars  each  paid. 


Therefore, 


x-8 
12        1^12 
jr  —  3     5      X 
Solving,  x=i  16  or  -  12. 

The  second  value  of  x  is  evidently  inadmissible,  since  there  could  not 
be  a  negative  number  of  persons.  Hence,  the  number  of  persons  in  the 
party  was  15. 

11.  A  chib  had  a  dinner  that  cost  $60.  If  there  had  been 
■'>  persons  more,  the  share  of  each  would  have  been  $1  less. 
I  low  many  persons  were  there  in  the  club  ? 

12.  A  party  of  young  people  agreed  to  pay  $8  for  a  sleigh 
ride.  As  4  were  obliged  to  be  absent,  the  cost  for  each  of  the 
1 1'st  was  10  i  greater.     How  many  went  on  the  ride  ? 

13.  A  tub  of  dairy  butter  weighed  20  pounds  less  than  a  tub 
of  creamery  butter,  and  360  pounds  of  dairy  butter  required  3 
more  tubs  than  the  same  amount  of  creamery  butter.  What 
weight  of  butter  was  there  in  a  tub  of  each  kind  ? 

14.  A  moving  picture  film  150  feet  long  is  made  up  of  a 
certain  number  of  individual  pictures.  If  these  pictures  were 
J  of  an  inch  longer  there  would  be  600  less  for  the  same 
length  of  film.     How  long  is  each  separate  picture  ? 

15.  Two  coats  of  paint  applied  to  the  sides  of  a  barn  having 
an  area  of  195  square  yards  required  69  pounds  of  paint.  One 
pound  covered  1.V  square  yards  more  for  the  second  coat  than 
for  the  first.  What  area  did  1  pound  of  paint  cover  for  each 
coat? 


300  QUADRATIC    EQUATIONS 

16.  A  purchase  of  80  four-inch  spikes  weighed  3  pounds  less 
than  one  of  80  five-inch  spikes.  If  1  pound  of  the  former  con- 
tained 6  spikes  more  than  1  pound  of  the  latter,  how  many  of 
each  kind  weighed  1  pound  ? 

17.  Mr.  Field  paid  $8.00  for  one  mile  of  No.  9  steel  wire 
and  $2.88  for  one  mile  of  No.  14  wire.  The  No.  9  wire 
weighed  224  pounds  more,  and  cost  i^  per  pound  less,  than 
the  No.  14  wire.     Find  the  cost  of  each  per  pound. 

18.  A  train  started  16  minutes  late,  but  finished  its  run  of 
120  miles  on  time  by  going  5  miles  per  hour  faster  than  usual. 
What  was  the  usual  rate  per  hour  ? 

19.  To  run  around  a  track  1320  feet  in  circumference  took 
one  man  5  seconds  less  time  than  it  took  another  who  ran  2 
feet  per  second  slower.     How  long  did  it  take  each  man  ? 

20.  Two  automobiles  went  a  distance  of  60  miles,  one 
making  6  miles  per  hour  faster  time  than  the  other  and 
completing  the  journey  f  of  an  hour  sooner.  How  long  was 
each  on  the  way  ? 

21.  A  cistern  can  be  filled  by  two  pipes  in  24  minutes.     If 

it  takes  the  smaller  pipe  20  minutes  longer  to  fill  the  cistern 

than  the  larger  pipe,  in  what  time  can  the  cistern  be  filled  by 

each  pipe  ? 

Solution 

Let  X  —  the  number  of  minutes  required  by  the  larger  pipe. 

Then,     a;  -f  20  =  the  number  of  minutes  required  by  the  smaller. 

Since  i  —  the  part  which  the  larger  pipe  fills  in  one  minute, 

X 

— - —  =:  the  part  which  the  smaller  pipe  fills  in  one  minute, 

x  +  20  ^ 

and  ^^  —  the  part  which  both  pipes  fill  in  one  minute, 

then,  ^  +  — ^- —  =  — .  . 

»•      r  +  20      24 

Solving,  rr  =  40  or  -  12. 

Hence,  the  larger  pipe  can  fill  the  cistern  in  40  minutes,  and  the  smaller 
pipe  in*^0  minutes. 


QUADRATIC   EQUATIONS  301 

22.  A  city  reservoir  can  be  filled  by  two  of  its  pumps  in 
.Ma\  >.  The  larger  pump  alone  would  take  1|  days  less  time 
than  the  smaller.     In  what  time  can  each  fill  the  reservoir? 

23.  A  company  owned  two  plants  that  together  made  25,200 
toncrete  building  blocks  in  12  days.  Working  alone,  one  plant 
would  have  required  7  days  more  time  than  the  other.  What 
was  the  daily  capacity  of  each  plant  ? 

24.  The  number  of  strawberry  baskets  made  by  a  machine 
was  12  more  per  minute  than  the  number  of  peach  baskets 
made  by  another  machine.  One  day  the  former  machine  started 
45  minutes  after  the  latter,  but  each  finished  2400  baskets  at 
the  same  instant.     Find  the  rate  of  each  per  minute. 

25.  Find  two  consecutive  integers  the  sum  of  whose  recip- 
rocals is  fjj. 

26.  Find  the  price  of  eggs,  when  2  less  for  30  cents  raises 
t  he  price  2  cents  per  dozen. 

27.  A  merchant  sold  a  hunting  coat  for  $11,  and  gained  a 
])er  cent  equal  to  the  number  of  dollars  the  coat  cost  him. 
What  was  his  per  cent  of  gain  ? 

28.  By  receiving  two  successive  discounts,  a  dealer  bought 
silverware  listed  at  $20  for  $9.  What  were  the  discounts  in 
per  cent,  if  the  first  was  5  times  the  second  ? 

29.  In  gathering  187^  bushels  of  tomatoes,  the  pickers  used 
baskets  that  held  ^  of  a  bushel  less  than  the  crates  in  which 
the  tomatoes  were  shipped.  The  number  of  basketfuls  picked 
was  50  more  than  the  number  of  crates  filled.  Find  the  capa- 
city of  a  basket ;  of  a  crate. 

30.  Each  page  of  a  book  of  400  pages  was  10  inches  by  f) 
inches.  The  publishers  decided  to  save  1550  square  inches  of 
paper  on  each  book  in  future  editions  by  cutting  down  the  mar- 
gin equally  on  every  side.  By  what  width  was  the  margin 
reduced  ? 

St-r,r,r«Tinv.  — The  number  of  leaves  in  each  book  =  |  the  number  of 


302 


QUADRATIC    I:QUATI0NS 


Formulae 

399.    1.    In  any  right-angled   triangle  (Fig.  1),  c- 
Find  all  the  sides  when  a  =  c  —  2  and  ^  =  c  —  4. 


a'  -f  b' 


Fig.  3. 

2.  The  area  (A)  of  a  triangle  (Fig.  2)  is  expressed  by  the 
formula  A  =  ^  ah.  If  the  altitude  (h)  of  a  triangle  is  2  inches 
greater  than  the  base  (a)  and  the  area  is  60  square  inches, 
what  is  the  length  of  the  base  ? 

3.  If  two  chords  intersect  in  a  circle,  as  shown  in  Fig.  3, 
a  •  ^  is  always  equal  to  c-d.  Compute  a  and  b  when  c  =  4, 
d  =  6,  and  b  =  a  -{-  5. 

4.  The  formula  h  =  a  +  vt  —  16^^  gives,  approximately,  the 
height  (h)  of  a  body  at  the  end  of  t  seconds,  if  it  is  thrown 
vertically  upward,  starting  with  a  velocity  of  v  feet  per  second 
from  a  position  a  feet  high. 

Solve  for  t,  and  find  how  long  it  will  take  a  skyrocket  to 
reach  a  height  of  796  feet,  if  it  starts  from  a  platform  12  feet 
high  with  an  initial  velocity  of  224  feet  per  second. 

6.  How  long  will  it  take  a  bullet  to  reach  a  height  of 
25,600  feet,  if  it  is  fired  verticall}^  upward  from  the  level  of 
the  ground  with  an  initial  velocity  of  1280  feet  per  second  ? 

6.  When  a  body  is  thrown  vertically  doivnicard,  an  approxi- 
mate formula  for  its  height  is  h  =  a  —  vt  —  'iGt-,  in  which  h,  a, 
1',  and  t  stand  for  the  same  elements  as  in  exercise  4. 

Solve  for  t,  and  find  when  a  ball  thrown  vertically  downward 
from  the  Eiffel  tower,  height  984  feet,  with  an  initial  velocity 
of  24  feet  per  second,  will  be  368  feet  above  ground. 

Find,  to  the  nearest  second,  when  it  will  reach  the  ground. 


QUAUKATIC    EQLATIONS  80ii 

EQUATIONS    IN    THE   QUADRATIC    FORM 

400.  An  equation  that  contains  but  two  puwers  of  an  un- 
known number  or  expression,  the  exponent  of  one  power  being 
twice  that  of  the  other,  as  a.r"  +  bx''  +  c  =  0,  in  which  n  repre- 
sents any  number,  is  in  the  quadratic  form. 

EXERCISES 

401.  1.    (liven  x*  +  (I  .r^  —  40  =  0,  to  tind  the  values  of  x. 

SOLTTION 

j-4  -I-  (}  a-i  _  40  =  0. 
Factoring,  (ar-^  -  4)(x^  +  10)  =  0. 

.-.  3-2  _4  ^0ora;2  +  10  =  0, 
M.l  r  =±  2  or  ±  V-TO. 

2.   Given  x^  —  x^  =  6,  to  find  the  values  of  x. 
KiK-^T  Solution 

.ri  _  a^l  =  6. 

( \)inpleting  the  square,        x^  —x*  +  (^)^  =  ^. 
Extracting  the  square  root,  xi  —  A  =  ±  |- 

.'.  x^=  3  or -2. 
Raising  to  the  fourth  power,  x  =  81  or  16. 

Skcond  Solution 

xi  -  X*  =  6. 
Let  X*  =  p,  then,  xi  =p2,  and  p*  —  p  =  6. 

...  p2_p_6  =  o. 
Factoring,  (P  -  ii)(P  +  2)  =  0. 

.•.p  =  3or  -2; 
that  is,  X*  =  3  or  —  2. 

Whence,  x  =  81  or  16. 


804                            QUADRATIC  P:QUATI0NS 

Solve  the  following  equations  : 

3.  x^-lSa^-]-m  =  (l  11.  x^-:^x^  =  -2. 

4.  x*-25x'-\-Ui  =  0.  12.  x^-x^  =  6. 

5.  x'-\Ha^  +  :V2  =  0.  13.  x-^2Vx  =  3. 

6.  3x'  +  5x--S  =  0.  14.  x^-2x^=3. 

7.  5x*-\-6x^~n=0.  15.  .1-^  =  10  .r*  -  9. 

8.  2.'c^-8a.'2-90  =  0.  16.  {x-Sf -^2(x-3)=3. 

9.  0^1-5.^^4-6  =  0.  17.  (a^ +1)24-4 (ar^ 4-1)  =45. 
10.  x^-\-Sx^-2S  =  0.  18.  (.7^-4)2 -3(a-'-4)  =  10. 


19.    Solve  the  equation  x  —  Ax^-\-Sx^  =  0. 

Solution 
Let  x^  =p,  then,  x^  —  p^,  and  a;  =  p^. 

Then,  F^  _  4p2  +  Sj^  =  0. 

Factoring,  ;>(  p"^  —  4  jo  +  3)  =0. 

Whence,  jo  =  0 

or  |)2  —  4  ;)  +  8  =  0. 

Factoring,  (/)  -  1  )(7)  -  8)=  0. 

Whence,  p  =  I  or  j5  =  3. 


That  is. 


:r^  =  0,  1,  or  3. 
.-.  ic  =  0,  1,  or  27. 


Solve : 

20.  ^  —  ^x  —  5  :^  =  0.  23.  5  x  =  x\^x  4-  6Vx. 

21.  a?-3a.-^4-2a;'^=0.  24.  3  a;  =  a-^/^4-2^^. 

22.  x-^2x^ —  3x^  =  0.  25.  2  .r  +  v^=  15a;V^. 


C^UADRATIC    KQL'ATIONS  ;iOi> 


26.    (riv«Mi  j-^     7  .r  -I-  \  y-     7  ./•+  1H=  24,  to  find  the  value 

.f  X. 

Solution 

r-i  _  7  3C  +  VW-  7  3^  +  18  =  24.  (1) 

Adding'  18,  y*  -  7  x  +  18  +  Vy^-  7  r  +  18  =  42.  (2) 

l*utp  for  (x-^-  7  y  +  18)^  and  p^  for  (ac*  -1  r  +  18).  (3) 

(4) 

(«) 
(7) 


'i'hen. 

p-^-\-p-  42  =  0. 

Solving, 

p  =  6  or  —  7. 

That  is. 

>/«*-7a;+18  =  6 

Va«-7«+18=-7. 

Squaring 

(«), 

052  _  7  X  +  18  =  36. 

Solving, 

X  =  «  or  -  2. 

Squaring' 

(7), 

0-^-  7  a!  +  18  =4lt. 

Solviiin. 

.r=  •  ±Wv::). 

Hence,  the  roots  of  (  1 )  are  ^  =  ^,  —  '^^  I  ±  i  V173. 

27.    Solve  the  equation  x  4-  2 Va;  +  3  =  21 . 


28.  Solve  x*-3a-  +  2Var*-3a-  +  6=  IS. 

29.  Solve  the  equation  af^  —  9  r^  -h  H  =  0. 

Solution 

3B6_1»3^  +  8  =  0.  (1) 

Factoring,  (a*  -  l)(a-»  -  8)  =  0.  (2) 

'|-lirivf..iv.  jr"-  1  =0.  (8) 

I  :e«_8  =  0.  (4) 

If  the  values  of  x  are  found  by  transposing  the  known  terms  in  (3)  and 
I)  and  then  extracting  the  cul)e  root  of  each  member,  only  one  value  of 
/  will  be  obtained  from  each  equation.  But  if  the  equations  are  factored, 
f/iree  values  of  z  are  obtained  for  each. 

Factoring  CH),  (x  -  \)(x^  +  x  +  I)  =0,  (6) 

and  (4),  (x-2)(X''  +  2;r  +  4)=0.  (6) 

Writing  each  factor  equal  to  zero,  and  solving,  we  have  : 

From(5),  x  =  l,  K-l+Jv^^),  U- ^  -  v'^=^)-  (7) 

From  (6),  x  =  2,  -  1  +  V^^,  -  1  -  V^^!.  (8) 

milne's  stand,  alo.  —  20 


306  QLADRATIC    EQUATIONS 

Note.  —  Since  the  values  of  x  in  (7)  are  obtained  by  factoring 
r^  —  1  =  0,  tliey  may  be  regarded  as  the  three  cube  roots  of  the  number 
1.     Also,  the  values  of  x  in  (8)  may  be  regarded  as  the  three  cnbe  roots 

of  the  nnmher  8  (§  286). 

Solve : 

30.    .c*' -  28  .r -f  1^7  =  0.  31.    a;^-16  =  U. 

32.  Find  the  three  cube  roots  of  —1. 

33.  Find  the  three  cnbe  roots  of  —  8. 

34.  Solve  the  equation  x^  +  4  x^  —  8  .v  +  3  =  0. 

First  Solutiox 
Extracting  the  square  root  of  the  first  member  as  far  as  possible, 

X*  +  4  x-^  _  8  .r  +  8  Ix^  +  2  X  -  2 


I  x^  +  2  X 

4x8 

4  x3  +  4  x-2 

2  X-  -f  4  X 

-  2       _  4  X-!  -  8  X  +  3 

-  4  x^  -  8  X  +  4 

-1 

Since  the  first  member  lacks  1  of  being  a  perfect  square,  the  square 
may  be  completed  by  adding  1  to  each  member,  which  gives  the  follow- 
ing equation :  ..or.,. 

"^  x4  +  4x^-8x+4=l. 

Extracting  the  square  root,     x2-i-2x  —  2=  ±1. 

.-.  x-i  +  2  X  -  8  =  0,  and  x"^  +  2  x  -  1  =  0. 
Solving,  X  =  1,  —  3,  -  1  ±  \/2. 

Second  Solution 

If  4x2  is  added  to  the  first  two  terms  of  the  given  equation,  the  re- 
sulting trinomial,  x*  4-  4  x^+  4  x^,  will  be  a  perfect  square.  Adding  4x2 
and  subtracting  4  x'^  does  not  change  the  first  member. 

Then,  x*  +  4  x^^  +  4  .x"^  -  4  x-  —  8  x  +  3  =  0  ; 

whence,  (x''^  +  2  x)^  —  4(x2  -f-  2  x)  +  3  =  0. 

Factoring,  (x'-^  -h  2  x  —  3)  (x^  +  2  x  -  1)  =  0. 

Solving,  X  =:  1,  —  3,  -  1  i  V2. 


QUADRATIC    EQUATIONS  307 


Thiku  Som  tion 
Hy  applying  the  factor  theorem  (§  104).  iln    ta(  tons  of  the  tirst  iiiem- 
Ijer  are  found  to  be  x  -  1,  x  +  8,  and  X''  +  2  x  -  1  ;  that  is, 

(X  -  l)(x  +  Z)(x-  +  2  X  -  1)  =  0. 

Solving,  X  =  1,  -  :J,  -  1  ±  >/2. 

Solve : 

35.  x'  +  23^-x  =  :^.  37.    j'*-2r'-\-.r  =  lS2. 

36.  X*--^jt'-\-Sx=-X  38.    x*  -(yx'-^27  x^lO. 

39.  X* -h  2  a^  +  5  it^ -^4  X -60  =  0. 

40.  .r*-f  6ar«  +  7a;--(;  ./--S^O. 

41.  ./•*-6ar'-|-ir).r-- l«u:  +  H  =  0. 

42      ^-^^•^±1=.-^1. 
'    x  +  1         x'        12 

SuGGERTiox.  — Since  the  second  term  is  the  reciprocal  of  the  first,  put 

p  for  the  first  terra  and  -  for  the  second. 

Then,  p+i  =  -. 

43.    ^±£^-2-  =2.  45.     "ili +  2(^4:4)  ^5_X. 

44      ?i±l+_t_=5.  .r^+l       4(0^-1)^21. 

4  ar'-fl      2  x-1         ar  + 1        5 

Solve  the  following  miscellaneous  equations  : 

47.  a:«-f-8ar^-0  =  0.  52.    x*  =  S  x-{-7  x^^x. 

48.  a?* -H  aci  —  2  =  0.  53.  «  — 5-f  2  Vx  —  5  =  8. 

49.  v^^+3\/x  =  30.  54.  x-f  10  =  2V«TlO  +  5. 

50.  aaf  4-  6x"  -I-  c  =  0.  55.  .,.  _  ;^  ==  21  -  4  \  .r^ll 

51.  .,.:_.,^'_i2a;i  =  0.  56.  L'./-.Sx  L:x-  +  5=-r). 


308  QUADRATIC    EQL\\T[()NS 

57.  2x-6V2x-  \  =H.  59.    af-\-xVx-T2  =  0. 

58.  it- =  11  —  3 ViC  HhT.  60.    x~^ —  ox~^ -^^  =  0. 


61.  .r-o.c-}-2V^-5a;-2  =  10. 

62.  xr  —  x—  V.^"^  —  ar+4  —  8  =  0. 


63.  a;--5a;  +  oVa^-5a;  +  l=49. 

64.  (x--2xy~2(;x'-2x)=S, 

65.  (0^2  -  ic)=^  -  (ar^  _  a;)  -  1.32  =  0. 


l^-lV  +  s/'— -1)=33. 


e..  ,.  +  lY_2(.-H^     ^ 


68     -•l+i^Y^->A+^' 


V        J  \       X 


70.  .f^- 10.*'"^ +  35  of' -50  if +  24  =  0. 

71.  1(5  x*-S  x^  -  31  x^-\-Sx-\-  15  =  0. 

72.  4ic*  — 4.r— 7.r-'  +  4a:  +  3  =  0. 

1  <S 

73.       .1-  +  X-+1   -  z=:-^ 

..•-  +  r  +  1       3 

74.  ar^_2.r+-^-^---^=4.        76.     .»^-.r+    ,,-^    --   =7 

x^  —  2x-\-i  .r  — .r^4       . 

75.  xF-Sx-^- ^--     =1.        77.         -^ ^  +  ■ -"^  =  -  ^-'^ 

1  '^ 

78.     ^ -^  +         _  ^_^^  -  ^i  =  0. 

l+aj  +  a;-      vl+.>^  +  a;- 


(^lADllATIC    EQUATIONS  'MV.) 

SIMULTANEOUS   EQUATIONS    INVOLVING    QUADRATICS 

402.  Two  simultaneous  quadratic  equations  involving  two 
unknown  numbers  generally  lead  to  equations  of  the  fourth 
degree,  and  therefore  they  cannot  be  solved  usually  by  quad- 
ratic methods. 

However,  there  are  some  simultaneous  equations  involving 
(piadratics  that  maybe  solved  by  quadratic  methods,  as  shown 
in  tlie  following  cases. 

403.  When  one  equation  is  simple  and  the  other  of  higher  degree. 

Equations  of  this  class  may  be  solved  by  finding  the  value 
of  one  unknown  number  in  terms  of  the  other  in  the  simple 
.<  I  nation,  and  tlieii  substituting  that  value  in  the  other  equation. 


404.    1.   Solve  the  equations  < 


EXERCISES 

=  43. 


SOLITTION 

x  +  y  =  7.  (1) 

.3a;2  +  y2=43.  (2) 

From  (1),  y  =  1  -  X.  (8) 

Substituting  in  (2),  3  z-«  +  (7  -  x)^  =  48.  (4) 

Solving.  r  =  3  or  i.  (6) 

Substituting  8  for  a;  in  (3),  y  =  4.  (6) 

Substituting  \  for  a?  in  (8),  y  =  y.  (7) 

f  when  r  =  a,  tj  =  4, 
'I'hat  i.s.  J-  and  v  eai-h  have  two  values  i 

[  wlien  a:  =  i,  .V  =  V- 


Solve  the  following  equations : 

'I 


\x-y  =  2. 


lOa-  4-  .V  =  3x-y,  r  m-  -  3  u'  =  13, 

V  —  J*  =  -.  I  ///  —  2  //  =  1 . 


310  QUADRATIC    EQUATIONS 

fx-  =  6-7/,  r3.'c(?/  +  1)  =  12, 

6.     -i  '  8. 

{ .rq{  X  -  2  v)  -  10,  f  3  rs  -  10  /•  =  s, 

U?/  =  10.  12-8=: -r. 

405.  An  equation  that  is  not  affected  by  interchanging  the 
unknown  numbers  involved  is  called  a  symmetrical  equation. 

2  x^  -\-  xy  +  2  y^  =  A  and  x'^  -f  ?/2  =  {)  are  symmetrical  equations. 

406.  When  both  equations  are  symmetrical. 

Though  equations  of  this  class  may  be  solved  usually  by 
substitution,  as  in  §§  403,  404,  it  is  preferable  to  find  values  for 
X  -f  y  and  x  —  y  and  then  solve  these  simple  equations  for  x 
and  y. 

EXERCISES 

407.  1.  Solve  the  equations  i  ^     ^~   ' 

^  \xy  =  l(). 

Solution 

X  +  ?/  =  7.  (1) 

xy  =  10.  (2) 

Squaring  (1),  a;^  +  2  xy  +  ?/2  =-^9.  (3) 

Multiplying  (2)  by  4,  Axy  =  40.  (4) 

Subtracting  (4)  from  (3),  x'^-2  xy  +  y^  =  9.  (6) 

Extracting  the  square  root,  x  —  y  =  ±  '^.  (6) 

From  (1)  +  (6),  x  =  5  or  2. 

From  (1)- (6),  y  =  2  or  6. 

f  oy^  -f-  w2  =  25 
2.    Solve  the  equations  \          '  ^ 

lx  +  y  =  l. 

Suggestion.  —  From  the  square  of  the  second  equation  subtract  the 
first  equation  ;  then  subtract  this  result  from  the  first  equation  and  pro- 
ceed as  in  exercise  1. 


QUADRATIC   EQUATIONS 

iiiatious  \ 

«"> 

Soil    1  ION 

x*  +  y*=97 

x+y  =  \. 

811 


Raising  (2)  to  the  fourth  p<»wer, 

X*  +  4  x'hj  +  «  x'Y  +  •*  ^y^  +  y*  =  1. 
Subtracting  (1)  from  (3),      4 x»y  +  6 x'Y  +  * ^V'  =  -  9«. 
Dividing  by  2,  2  xy  +  8  x^  +  ^  «y'^  =  -  ^^• 

Ixy  X  Wjuare  of  (2).  _'  r^y  +  4  x-y-  +  2  xy*  =  2xy. 

Subtracting  (ft)  from  ^«i;,  x'^y*  —  2  xy  =  48. 

Solving  for  xy,  xy  =  —  6  or  8. 

Equations  (2)  and  (8)  give  two  pairs  of  «iinultaneou8  equations, 
f  X  4-  y  =  1       ,  I  .'•  +  y  =  1 

.'  and 

I  xy  =      »;  I  xy  =  8 

Solving  as  in  exercise  1,  the  corresponding  values  of  x  and  y  are 

|X=      3;  -2;  Ul+>/-^i);  i  {\  -  \/-'^') '^ 


b=-2;       3;  \i^-^^- 

Solve  the  following  equations : 

^     jar  4- .y-  =  50, 

I  x;v  =  7. 


p4-y  =  y. 
ljr»-|-V»  =  243. 

ra:^  +  a:y +  2^^  =  31, 
1. ,-'  +  2^=26. 


31); 


6. 


7. 


10. 


11. 


^(i+V-si). 

X-  -  u;j/  -h  /  =  4. 

I  ^'4-/  =  13, 
I  .c  -|-  //  -I-  XV  =  1 1 . 


0) 

(2) 

(3) 
(4) 
(5) 
(«) 
(7) 
(8) 


cy  =  6. 


1  .cy  = 

f  X* -!-,/-•=  1()0, 
12.  • 


13. 


1x^^=^  =  144. 


312  QUADRATIC    EQUATI(3NS 


14. 


[x  +  y  =  ?y.  1  ic^  -f  .17/  +  f  =  7. 


408.  An  equation  all  of  whose  terms  are  of  the  same  degree 
with  respect  to  the  unknown  numbers  is  called  a  homogeneous 
equation. 

x:^  —  xy  =  y^  and  Sx^  +  y'^  =  0  are  homogeneous  equations. 
An  equation  like  x^  —  xy  -\-y-  =  21  is  said  to  be  homogeneous 
in  the  unknown  terms. 

409.  When  hoth  equations  are  quadratic,  one  being  homogeneous. 

In  this  case  elimination  may  always  be  effected  by  substitu- 
tion, for  by  dividing  the  homogeneous  equation  through  by  2/^  it 

becomes  a  quadratic  in  -  •    The  two  values  of  -  obtained  from 

y  y 

this  equation  give  two  simple  equations  in  x  and  y,  each  of 
which  may  be  combined  with  the  remaining  quadratic  equa- 
tion as  in  §§  403,  404. 

Thus,  aa^ -i- bxy -\- cy^  =  0  is  the  general  form  of  the  homo- 
geneous equation  in  which  a,  b,  and  c  are  known  numbers. 

//^»\  *         /HrK  or 

Dividing  by  y^,  we  have  af  -  j  -{-b\~]-\-G  =  0,3L  quadratic  in  -  • 

EXERCISES 

410.  1.   Solve  the  equations   ]  ^^  "^^      .V  —    » 

[  Ty  x^  -^  4:  xy  —  if  =  0. 

Solution 

a;2  +  8x— </  =  5.  (1) 

5ic-^  +  4.x«/- «/-^  =  0.  (2) 

Dividing  (2)  by  y^,  5[ -V  -|-  4/^')  -  1  =  0,  a  quadratic  in  -  which  may 

be  solved  by  factoring  or  by  completing  the  square. 

To  avoid  fractions,  however,  (2)  may  be  factored  at  once ;  thus, 

(x-hy)(^x-y)  =  0. 
.'.  y  =  —  X  or   5 «. 


QUADKATK     KQUATFONS  313 

Suh8titutin<r  —  x  for  //  in  ri).  simplifying,  etc., 

.,■•-'  -f  4  y  =  5. 
Solving.  J?  =  1    or    -  5.  (8) 

.-.  y  =  -  05  =  -  1    or    6.  (4) 

Substituting  '\x  for  y  in  d),  simplifying,  etc., 

X-  -  2  X  =  ■). 
Solving,  :>•  =  1  -I-  \/($  or  1  -  V6.  (6) 

.-.y  =  .-,x  =  5(l+\/6)or6(l-v/«).    ((5) 

Hence,  from  (3),  (4),  (6),  and  (0)  tlie  roots  of  the  given  equation  are 

I  a;  =  1  :  -  6  ;  1  +  >/6  ;  1  -  Vd  ; 

lj/  =  -l;  5;  6(1+Vtf);  5(1 -\/«). 

Solve  the  following  equations : 

6  a:^  —  o  an/ —  () /y- =  0.  i  ur  — xt/ —  122/^  =  8. 


^     1 5ar' 4-8i:2/-4t/^  =  0,  ^  r^_ary-/  =  20, 

la^  +  2/=:60.  *  l3ar'-13an/  +  12r'  =  0. 

^      r2a^-X2/-V  =  0,  g  r3a^-  7.r//  +  4/  =  0, 

l4a:*4-4iP2/  +  /r  =  .'^n.  '  I  5  a:^.  7  j^_^  3^y2^  4^ 

^      M;.r  +  .r//-lL'/r  =  0,  ^  |  ar^ -h?/4-x-.v  =  12, 


411.  When  both  equations  are  quadratic  and  homogeneous  in 
the  unknown  terms. 

Either  of  the  following  methods  may  be  employe<l  in  this 

I  so  : 

Substitute  vy  for  *,  solve  for  y*  in  each  equation,  and  com- 
pare the  values  of  jy*  thus  found,  forming  a  (luadratic  in  v. 

Or,  eliminate  the  absolute  term,  forming  a  homofjeneous 
(■(juation;  then  proceed  as  in  §§  409,  410. 


314  QUADRATIC    EQUATION'S 


EXERCISES 


412.    1.  Solve  the  equations  I '^''"    •^•'/  +  -^'- -1= 

\y--2xti=-\r>. 


.  (1) 
(2) 
(•^) 
(4) 
(5) 
(6) 

(7) 

2v  -  1      1)-^- v+  1  ^^^ 

Clearing,  etc.,                  5  v"^  -  19  v  +  12  =  0.  (0) 

Factoring,                         {v  -  8)  (5  ?;  -  4)  =0.  (10) 

.-.  V  =  3or  ;t.  (11) 


\y--^xji=  —u 

FlKST    SOLl-TION 

x-'-xy  +  y^  =  2\. 

y^-2xy=  -  15. 

Assume 

X  =  vy. 

Substituting  in  (1), 

vY  -  vii^  +  y^  =  2\. 

Substituting  in  (2), 

y'  -2vy^=  -15. 

Solving  (4)  for  y\ 

.2  _         21 

v^-v+l 

Solving  (5)  for  y^. 

2i;-l 

nnmnnrina  tViP  valnps 

15      _         21 

Substituting  3  for  v  in  (7)  or  in  (6),     y  = 
and  since  x  =  vy,  x  == 


±V8    I 
±8V3.j 

=  ±5, 

=  ±4.j 


(12) 
(13) 


Hence, 


Substituting  4  for  v  in  (7)  or  in  (0),      i/  =  ±  5 
and  since  x  =  vy,  x 

When  the  double  sign  is  used,  as  in  (12)  and  in  (13),  it  is  understood 
that  the  roots  shall  be  associated  by  taking  the  upper  signs  together  and 
the  lower  signs  together. 

a-  =  3V3:         -  3V3;         4;  -4; 
y  =     V.S  ;  -  VS;  5  ;   -  5. 

Suggestion  for  Second  Solution.  —  Multiplying  the  first  equation  by 
5  and  the  second  by  7,  and  adding  the  results,  we  eliminate  the  absolute 
term  and  obtain  the  homogeneous  equation 

^x^-  I9xy+  12  2/2  =  0, 
which  may  be  solved  with  either  of  the  given  equations,  as  in  exercise  1, 
§410. 

Solve  the  following  equations  : 
^      (xy^3f  =  20,  3     ^x^-\-xy  =  12, 

[x^-3xy=  -H.  \xy  +  2y'  =  b. 


6. 


QL'ADRATIC    EQUATION'S  :\U) 

i  ^^  —  ]r  =  3.  [X-  -  ]r  —  To. 

I  .r'  -  ^/  -  y-  =  20,  ,  J-  _  ,-i  .,•//  +  3  y  =  8, 

1  jr  -  :5  xtf  +  2  //-  =  8.  ■     I  .S  .^  -|_  xt/  +  ?/■'  =  1-^4 . 

413.    Special  devices. 

Many  systems  of  simultaneous  equations  that  belong  to  one 
<»r  more  of  the  preceding  classes,  and  many  that  belong  to  none 
of  them,  may  be  solved  readily  by  special  devices.  It  is  impos- 
sible to  lay  down  any  fixed  line  of  procedure,  but  the  object 
often  aimed  at  is  to  find  values  for  any  two  of  the  expressions, 
X  -{•  y,  X  —  y,  and  xy,  from  which  the  values  of  x  and  y  may  be 
obtained.  Various  manipulations  are  resorted  to  in  attaining 
this  object,  according  to  the  forms  of  the  given  equations. 

EXERCISES 


414.    1.    Solve  the  equations 


f  x'  +  xy  =  12, 
\xy-\-f==A, 


Solution 

x2  +  xy  =  12.  (1) 

xy+y^  =  A.  (2) 

Adding  (1 )  and  (2),        x=^  +  2  xy  +  y*  =  16.  (J^) 

:.  X  +  y  =  +  4  or  -.4.  (4) 

Subtracting  (2)  from  (1),  x^  -  y-^  =  8.  (6) 

Dividing  (6)  by  (4),  x  -  y  =  +  2  or  -  2.  (6) 

Combining  (4)  and  (6),  x  =  3  or  —  3  ;  y  =  1  or  —  1 . 

Note.  —  The  first  value  of  x  —  y  corresponds  only  to  the  first  value  of 
y,  and  the  second  value  only  to  the  second  value. 

Consequently^  there  are  only  two  pairs  of  values  ofx  and  y. 

Ob.serve  that  the  given  equations  belong  to  the  class  treated  in  §  411. 
I'he  special  device  adopted  here,  however,  gives  a  much  neater  and  simpler 
olution  than  either  of  the  methods  presented  in  that  case. 


816  (illADKATIC    KQL'A'L'IOXS 

2.    Solve  the  equations    |  •«'' +  r +  .^• +  // =  U, 
I  xy  =  3. 

Solution 

x2  +  y2  +  a._^y^l4,  (1) 

xy  =  8.  '  (2) 

Adding  twice  tlie  second  equation  to  the  first, 

.1-^  +  2  ary  +  y^  +  J-  -h  y  =  20. 
Completing  the  square,   (a-  +  //)^  +  (x  +  y)  -\-(hy^  =  20\. 
Extracting  the  square  root,  x  +  y  -\-  ^  =  ±  |. 

.".  X  +  y  =  4  or  —  5.         ■  (8) 

Equations  (2)  and  (3)  give  two  pairs  of  simultaneous  equations, 

'  x  +  y  =  4        ,    (  X  +  y  =  -  6 
and  \ 
xy  =  ?>  {  xy  =  8 

Solving,  the  corresponding  values  of  x  and  y  are  found  to  be 

(x  =  S;    1;    i(-5+Vl3);    |(  -  o  -  VlS)  ; 
|y=l;   8;    |(-5-Vi3).    i(-_5+Vl3). 

Symmetrical  except  as  to  sign. — When  one  of  the  equations 
is  symmetrical  and  the  other  v^ould  be  symmetrical  if  one  or 
more  of  its  signs  were  changed,  or  when  both  equations  are  of 
the  latter  type,  the  system  may  be  solved  by  the  methods  used 
for  symmetrical  equations  (§  406). 


3.    Solve  the  equations 


x^  +  f  =  53, 


Ix 

-2/  =  5. 

Solution 

X'  +  2/2 

=  53. 

X  -y 

=    5. 

c-^- 

2  xy  +  y^ 

=  25. 

'Ixy 

=  28. 

J'^  + 

2  xy  +  y^ 

=  81. 

y  +  y 

=  ±9. 

X 

=  7  or 

y 

=  2  or 

0) 

(2) 

Sciuaring  (2),                           x-  —  'lxy  +  y~  =  25.  (3) 

Subtracting  (8)  from  (1),                        '2xy  =  28.  (4) 

Adding  (4)  and  (1),              x'^  +  '2xy  +  ?/''  =  81.  (5) 

Extracting  the  square  root,                 x  +  ?/  =  ±  9.  (6) 
Fropi  (6)  and  (2), 
and 


QUADRATIC    EQUATIONS  31^ 


4.   Solve  the  equations  . 


Jr     y- 


KX       y 
SuooBSTios.  —  Square  the  second  equation  and  im>cee(l  exactly  as  in 
>  icise  3.     Solving  at  first  for      and  -  instead  >£  lor  x  and  y,  the  result  is 

0) 

and  -  =  5  or  —  7.  (2) 

Solving  (1), 
Solving  (2), 

XoTK.  —  It  is  sometimes  convenient  to  begin  by  solving  for  expressions 
"ilier  than  x  and  y,  as  Vacy,  y/x-\-  y,  etc. 

VV^hether  the  equations  are  symmetrical  or  symmetrical  ex- 
cept for  the  sign,  it  is  often  advantageous  to  substitute  u-\-v 

f<  )r  x,  and  n  —  r  for  //. 


J- 

X 

y 

-6, 

-  =  5or 

y 

-7. 

ar  =  1  or 

-i- 

y  =  ^  or 

-f. 

5.    Solve  the  equations 


.r  —  //  =  U. 


SoLHTION 

a:*-|-y*  =  82.  (1) 

x-y  =  2.  (2) 

Assume                                           x  =  ti  -I-  r,  (3) 

and                                                      y  =  u  —  v.  (4) 

Substituting  these  values  in  (1), 

u*  +  4  w«r  4-  ('}  n'^vt^  +  4  uu '  +  v* 

4-  M«  -  4  m'o  +  (J  ?«Vi  -  4  ttr «  +  r^  =  82,  (6) 

and  in  (2),                                           2r=2.  (6) 

Dividing  (6)  by  2,                u*  +  6  ?««»*  +  tr*  =  41.  (7) 

Dividing  (6)  by  2,  t)=l.  (8) 

Substituting  1  for  r  in  (7)  and  solving,    u  =  i  2  or  ±  V-  10.        <0) 

Hence,  8nb8titMtin^  (8)  and  (9)   in   (3)  and    (4^.   tlie  corresponding 
values  of  X  and  y  are  found  to  be 

J   r  =  3;          -1;  i    ;    >/- 10 ;  1_V_10; 

I  y  =  1 ;  -  1  +  v/:rio ;       -  i  -  V^^^. 

Note.  —  The  givin  .s\>irm  of  equations  may  be    snlvpd  also    by  the 
Tiitthcxi  of  exercise  3,  §  407. 


818  QUADRATIC    EQUATIONS 

Division  of  one  equation  by  the  other.  —  The  reduction  of  equa- 
tions of  higher  degree  to  quadratics  is  often  effected  by  divid- 
ing one  of  the  given  equations  by  the  other,  niemher  by  member. 


6.    Solve  the  equations 

(-.(•■'  +  . >•-/  +  //' =  91, 
I  .y- —  .r  V -f- //- =  7. 

Solution 

X^  +  X^,/  Jr  ?/  =  '.>!. 

(1) 

x~  -xy  -\-  y^  =  1. 

(2) 

Dividing  (1)  by  (2), 

x'  +  xy-{-y^=lS.    * 

(3) 

Subtracting  (2)  from  (3), 

2xy=:6; 

whence, 

xy  =  3. 

(^) 

Adding  (4)  and  (3). 

xi^2xy  +  y^  =  H^. 

(5) 

Subtracting  (4)  from  (2), 

x:^  --2xy  +  ?/  =  4. 

(6) 

Extracting  the  square  root  of  (5) .             x  +  ?/  =  4  or  -  4. 

0) 

Extracting  the  square  root  of  (0),            x  —  y  —  2  or  —  2. 

(8) 

Solving  these  simultaneous 

equations  in  (7)  and  (8), 

{X: 

=  3  ;   1  ;   -  1  ;    -  3  ; 

[v- 

=  1 ;  3  ;   -  3  ;   -  1. 

XoTE.  —  Since  (7)  and  (8)  have  been  derived  independently,  with  the 
first  value  of  x  +  y  we  associate  each  value  of  x  —  y  in  succession,  and 
with  the  second  value  of  x+  y  each  value  of  x  —  y  in  succession,  in  the 
same  order. 

Consequently,  there  are  four  pairs  of  values  of  x  and  y. 

( x'^'  _  jfi  =  26 
7.    Solve  the  equations    T        "  ' 

[x-!/  =  2. 

Suggestion.  —  Dividing  the  first  equation  by  the  second, 
.r-2  -L  xy  +  y-  =  13. 

Therefore,  solve  the  system, 

I  x-  +  xy  +  y-  =  13, 

\x-y=2, 
instead  of  the  given  system. 

Note. — It  is  sometimes  the  case  that  a  root  is  removed  when  one 
equation  is  divided  by  the  other,  member  by  member. 

Observe  that  the  given  system  may  be  solved  by  the  method  used  in 
exercise  5,  but  the  solution  suggested  here  is  briefer  and  simpler. 


(QUADRATIC    EQUATIONS  819 

Elimination  of  similar  terms.  —  When  the  equations  are  qiiad- 

tic  and  each  is  liouiogeneous  except  for  one  term,  if  these 

A(e]>te(l  terms  are  similar  in  the  two  equations,  they  may  be 

t'liiiiinaird  and  the  solution  of  the  system  be  made  to  depend 

oil  the  case  of  §  409. 

Some  equations  beloiii^ing  to  this  class,  nanieiy,  those  that  are  homo- 
neons  except  for  the  ah.solnte  term,  have  l)een  treated  in  §  411. 


.SolVf   tllf 


equations 

I  2 .»-  -  .n/  4-  »/^  =  2  V. 


Suggestion.  —  Multiplying  the  first  equation  by  4  and  the  second  by  6, 

4.r2-f-8a^  =  10y,  (a) 

.  1  10  X'  -  '>  xy  -\-by^-\0  y.  (6) 

Subtractiniit  («)  fro™  (*)' 

(\  x^  —  18  xy  -H  5  y'^  =  0,  a  homogeneous  equation. 

Therefore,  solve  the  system, 

Oa-«-  13rv  +  6y«  =  0, 
2  X-  -  xy  +  y-  -  2  y, 
instead  of  the  given  system,  using  the  method  of  §  409. 

Instead  of  eliminating  terms  below  the  second  degree,  as  in 
t'xercise  8,  in  certain  systems  it  is  advantageous  to  eliminate 
similar  terms  of  the  second  degree. 


9.    Solvt'  the  equations 


{xii-\-x  =  .So, 


Solution 

xy-\-x  =  Zb.  (1) 

xy  +  y  =  32.  (2) 

Subtracting  (2)  from  (1),  x-y  =  3; 

whence,  y  =  x  —  3.  (8) 

Substituting  (3)  in  (1),      x(x  -  3)  +  x  =  36, 
or  x«-2x  =  35. 

Solving,  X  =  7  or  —  5.  (4) 

Substituting  (4)  in  (3),  y  =  4  or  -  8. 


rVIi)  QL'ADKA'nC    KQL'A'i'lONS 


Solve,  using  tlie  methods  illustrated  in  exercises  1-9 : 

10.    1^^'+^'*''=^'  15      j^  c* -\- c'rf- -{- cl' : 
[  mn  -\-n-—~\.  I  c-  —  rd  -\-  d-  = 

...      hr  +  g=^  +  ^7  +  7  =  :^(>, 

Ux/=-lo.  16.  ' 


r  —  8  =  (>. 


12. 


rr'+6-=l;)0, 


I  .»•■      //- 

13.  ;  (xy  +  x  =  '^2, 
111,  18.    ^     -^  ^ 

I —  J-.  I  xii  +  w  =  27. 

[  X      y  J   ^  J 

14.  J       ^'  '                                   19.     •                  ./ 


415.  All  the  solutions  in  §§  403-414  are  but  illustrations  of 
methods  that  are  important  because  they  are  often  applicable. 
The  student  is  urged  to  use  his  ingenuity  in  devising  other 
methods  or  modifications  of  these  whenever  the  given  system 
does  not  yield  readily  to  the  devices  illustrated,  or  whenever 
a  simpler  solution  would  result. 

EXERCISES 

416.  Solve  the  following  miscellaneous  systems  of  equations  : 

'   \xy  =  2.  ■    U  +  a6  +  40  =  0. 

r5x2-4r'  =  44,  g     |.i'--^-3.x7/  =  8.r, 

Ua;2_5.v^'  =  19.  "    l2ar'-.i7/  +  .v-  =  8.f. 

3     |l  +  a^  =  .V,.  ^     fa-^4-ary-f./=21. 


.^•2  4-  ?y2  =  ()1 .  \  X-  —  xy 

jf-  -  xy  =  48,  r  X-  4-  ?/'  +  X 

8.  i 

.Ty-7/^'  =  12.  l.»rv=-12. 


quadkatk: 


10. 


11, 


13. 


14. 


15. 


16. 


IS. 


19. 


21. 


f .,-'  +  / =  40, 
I  u-^  -h  x}i  =  -  (), 

.r^  +  y  =  17, 
./•  —  //  =  —  3. 

\r?/4-  .r  =  44, 
.r7/  +  y-'=-2S. 

jr'-|-4ar-|-.S?/=  -1, 
2a:*  +  5a7/ 4-2^^  =  0. 


m      n 


2' 


1 
18 


0. 


=  6, 

:61. 

.  xy  -  f  = 

=  77, 

=  12. 

2x-y  = 

_  o 

.2x^  +  f 

=  ]• 

x'-hxy  +  f  = 

=  19, 

W-f  = 

rl9. 

or  +  Sxy 

=  !r 

+  23, 

x-\-^y=: 

:9. 

22. 


321 


23. 


24. 


25. 


26. 


27. 


28. 


29. 


30. 


31. 


32. 


33. 


34. 


f4.r//  =  9(>-jry, 

[x-hy  =  iK 

fx'-xy^H, 
\xy-\-y' =  12. 

(x(x-^y)  =  x, 

l.y(-c-.v)  =  -1. 

(a^-hSxy-y'  =  4S, 
[a;4-2y  =  10. 

\2x'-^3xy  +  f  =  2(), 
l5a^  4-4^  =  41. 

\2xy-y^=rj, 
3xy-f.5x'=10-i, 

jx'  +  xy-\-f=iryl, 
[x^-{-y'=106, 

(l+x  =  y, 


1  -har'  = 


.'A' 


[x'-f=<). 

\x'  +  xy-\.f-  =  S4, 
[x  —  ^/Qey-\-y=(). 

[4«*-2icy4-y'  =  13, 
i8ar^4-y*  =  65. 

6x^-\-6f  =  13xy, 
^x^-y^  =  20. 

V4-y^-3(2J4-.v)  =  8, 

ar4-.y  4-«y  =  ll. 


mii.ne's  stanh.  .\i.«;.  —  21 


322 


q  V  A 1 )  1 1  AT  1  c  yji  i:  a  no  x  s 


35. 

■ar^-7^^  =  ;^7, 

[xy(y-x)=-12. 

39. 

(3xy-\-2x  +  !J  =  2r>, 
'9af-4y-  =  0. 

36. 

'x  +  y  =  25, 

40. 

'X--7  xy  4- 12  //-  =  0, 
xy  +  3y  =  2x-\-21. 

37. 

(x'Jrf  =  225y, 
,.x-2-2/'  =  75. 

41. 

((x  +  y)(x'  +  f)  =  65, 
'Xx-y)(x^-y^)  =  rK 

38. 

r.>''  +  .r  =  3.^7/4-5, 

42. 

'x-  +  y  =  x  —  y-  +  A2, 
xy  =  20. 

43. 

'x-\-y-{-2^x 

+  !/  = 

=  24, 

.x  —  y-\-3\/x 

-I/  = 

=  10. 

44. 


45. 


^  +  2/^  +  6  Vx-  +  ?/2  =  55, 
x^  —  y-^i. 

x^  -  6  a;2/  +  9  /  4-  2  .T  -  6  ?/  -  8  =  0, 

Suggestion.  —  The  equations  may  be  written  in  the  quadratic  form 

(x  -  .3  yY  +  2(r^  _  3  ?/)  -  8  =  0, 
(a;  +  2  xjY  -  4(x  +  2  ?/)  -  21  =  0. 

f  if^  —  ic?/  =  a-  +  6- 


Thus, 


46.    Solve   \ "       "^  .       j^  £qj,  ^  ^^^  y^ 

{ xy  —  y-  =  2  ah 


x-2y  =  2(a-\-b) 


47.    Solve  \  ^         \     .   -/         1^  £^^.  ^  ^^^ 

\xy  +  2y^  =  2h{h-a) 


48.    Solve  \  ^  \  for  x  and  2/. 

x^y  -\-  xy'  =  2  a  (a^  —  h^) 


49.    Solve  J         '       \  for  a  and  t. 

\v  =  at     J 


50.    Solve  \  \  for  v  and  f. 

\^v  =  at  J 


QUADRATIC   EQUATIONS  828 

Problems 

417.  1.  The  sum  of  two  iimnbcis  is  \'J.  ami  tlieir  jji-odiict  is 
:'>-.     What  are  the  numbers? 

2.  The  sum  of  two  numbers  is  17,  and  the  sum  uf  their 
s(iuares  is  157.     What  are  tlie  numbers? 

3.  The  difference  of  two  numbers  is  1,  and  the  difference  of 
their  cubes  is  91.     What  are  the  numbers? 

4.  The  sum  of  two  numbers  is  82,  and  the  sum  of  their 
«iuare  roots  is  10.     What  are  the  numbers? 

5.  It  takes  52  rods  of  fence  to  inclose  a  rectangular  garden 
iitaining  1  acre     How  long  and  how  wide  is  the  garden? 

6.  The  product  of  two  numbers  is  59  greater  than  their  sum. 
and  the  sum  of  their  squares  is  170.     What  are  the  numbers? 

7.  If  63  is  subtracted  from  a  certain  number  expressed  by 
two  digits,  its  digits  will  be  transposed;  and  if  the  number  is 
multiplied  by  the  sum  of  its  digits,  the  product  will  be  729. 
What  is  the  number? 

8.  A  man  expended  $  G.OO  for  canvas.  Had  it  cost  4  cents 
less  jier  yard,  he  would  have  I'eceived  5  yards  more.  How 
many  yards  did  he  buy,  and  at  what  price  per  yard? 

9.  Mr.  Fuller  paid  $  2.25  for  some  Italian  olive  oil,  and  $  2.00 
a-  ^  gallon  less  of  French  olive  oil,  which  cost  $.50  more  per 
illon.     How  much  of  each  kind  did  he  buy  and  at  what  price? 

10.  In  papering  a  room,  18  yards  of  border  were  required, 
while  40  yards  of  paj^er  ^  yard  wide  were  needed  to  cover 
the  ceiling  exactly.     Find  the  length  and  breadth  of  the  room. 

11.  An  Illinois  farmer  raised  broom  corn  and  pressed  the 
<>120  pounds  of  brush  into  bales.  If  he  had  made  each  bale 
'_'()  poimds  heavier,  he  would  have  had  1  bale  less.  How  many 
bales  did  he  press  and  what  was  the  weight  of  each? 

12.  The  total  area  of  a  window  screen  whose  length  is  4  inches 
:^neater  than  its  width  is  10  square  feet.  The  area  inside  the 
wooden  frame  is  8  square  feet.     Find  the  width  of  the  frame. 


324  QUADRATTC   EQUATIONS 

rs.  A  boy  has  a  large  blotter,  4  inches  longer  than  it  is  wide, 
and  480  square  inches  in  area.  He  wishes  to  cut  away  enough 
to  leave  a  square  256  square  inches  in  area.  How  many  inches 
must  he  cut  from  the  length  and  from  the  width  ? 

14.  A  man  bought  4  more  loads  of  sand  than  of  gravel,  pay- 
ing $  .50  less  per  load  for  sand  than  for  gravel.  The  sand  cost 
him  $9.00  and  the  gravel  $10.00.  What  quantities  of  each 
did  he  buy  ?     What  prices  did  he  pay  ? 

15.  The  course  for  a  36-mile  yacht  race  is  the  perimeter  of  a 
right  triangle,  one  leg  of  which  is  3  miles  longer  than  the  other. 
How  long  is  each  side  of  the  course  ? 

16.  A  rectangular  skating  rink  together  with  a  platform 
around  it  25  feet  wide  covered  37,500  square  feet  of  ground. 
The  area  of  the  platform  was  J  the  area  of  the  rink.  AVhat 
were  the  dimensions  of  the  rink  ? 

17.  The  cubic  contents  of  a  produce  car  33  feet  long  were 
1848  cubic  feet,  while  the  cubic  contents  of  a  furniture  car  3 
feet  longer,  ^  foot  wider,  and  1  foot  higher  were  2448  cubic 
feet.  What  were  the  dimensions  of  each  car,  if  in  each  case 
the  width  exceeded  the  height  ? 

18.  Two  men  working  together  can  complete  a  piece  of  work 
in  6 1  days.  If  it  would  take  one  man  3  days  longer  than 
the  other  to  do  the  work  alone,  in  how  many  days  can  each  man 
do  the  work  alone  ? 

19.  After  a  mowing  machine  had  made  the  circuit  of  a 
7-acre  rectangular  hay  field  11  times,  cutting  a  swath  6  feet 
wide  each  time,  4  acres  of  grass  were  still  standing.  How 
long  and  how  wide  was  the  field? 

20.  The  fore  wheel  of  a  carriage  makes  12  revolutions  more 
than  the  hind  wheel  in  going  240  yards.  If  the  circum- 
ference of  each  wheel  were  1  yard  greater,  the  fore  wheel 
would  make  8  revolutions  more  than  the  hind  wheel  in  going 
240  yards.     What  is  the  circumference  of  each  wheel  ? 


QUADRATIC   EQUATIONS  82o 

21.  A  man  loaned  i^lOOO  in  two  unequal  sums  at  such 
rates  that  both  sums  yielded  the  same  annual  interest.  The 
larger  sum  at  the  higher  rate  of  interest  would  have  yielded 
$  36  annually,  the  smaller  sum  at  the  lower  rate,  $  16 
;iiinually.  What  sums  did  he  invest,  and  at  what  rates  of 
interest? 

22.  A  sum  of  money  on  interest  for  one  year  at  a  certain  per 
lit  amounted  to  $11,130.     If  the  rate  had  been  1  %  less  and 

1  lie  principal  $  100  more,  the  amount  would  have  been  the  same. 
I'ind  the  principal  and  rate. 

23.  The  town  A  is  on  a  lake  and  12  miles  from  B,  which  is 
I  miles  from  the  opposite  shore.     A  man  rows  across  the  lake 

:i(l  walks  to  B  in  3  hours.     In  returning,  he  walks  at  the 
line  rate  as  before,  but  rows  2  miles  an  hour  less  than  before. 

1  {■  it  takes  him  5  hours  to  return,  find  his  rates  of  rowing  and 

his  rate  of  walking. 

24.  A,  B,  and  C  started  together  to  ride  a  certain  distance. 
A  and  C  rode  the  whole  distance  at  uniform  rates,  A  2 
miles  an  hour  faster  than  C.  B  rode  with  C  for  20  miles, 
:iiid  then,  by  increasing  his  speed  2  miles  an  hour,  reached 
his  destination  40  minutes  earlier  than  C  and  20  minutes 
later  than  A.  Find  the  distance  and  the  rate  at  which  eac-h 
1  raveled. 

25.  The  distance  a  body  will  fall  in  t  seconds,  starting  from 
1  tst,  is  given  by  the  formula  8  =  ^gt^.  A  man  dropped  a  torpedo 
i  rom  a  height  and  heard  the  report  5  seconds  later.  Taking 
'/  =  32.16  and  the  velocity  of  sound  1125.6  feet  per  second,  find, 
lo  the  nearest  tenth  of  a  second,  the  time  durini:^  which  the 
lorpedo  was  falling. 

26.  A  mixture  of  graphite  and  clay,  to  be  used  as  "lead"  in 
]  lencils,  was  c  %  clay  and  weighed  p  pounds.  After  the  addition 
of  clay  to  make  the  "  lead  "  harder,  the  mixture  was  (c  +  10)% 
clay  and  weighed  240  pounds.  If  graphite  had  been  added, 
instead  of  clay,  until  the  mixture  weighed  250  pounds,  the 
mixture  would  have  been  (c  —  8)%  clay.     Solve  for  p  and  for  c. 


GRAPHIC   SOLUTIONS 


QUADRATIC  EQUATIONS 

418.    Graphic  solution  of  quadratic  equations  in  jr. 

Let  it  be  required  to  solve  graphically,  x-  —  6  a?  +  5  =  0. 

To  solve  the  equation  graphically,  we  must  first  draw  the 
graph  of  x~  —  6x  -{-  5.     To  do  this,  let  y  =  x^  —  6x  -\-  5. 

The  graph  of  y  =  x^  —  6x-{-5  will  represent  all  the  corre- 
sponding real  values  of  x  and  oi  xr  —  6  x  -\-  5,  and  among  them 
will  be  the  values  of  x  that  make  x^  —  6x  -\-  o  equal  to  zero, 
that  is,  the  roots  of  the  equation  cc^  —  6  oj  -f  5  =  0. 

In  substituting  values  of  x,  when  the  coefficient  of  x-  is  -|-  1, 
as  in  this  instance,  it  is  convenient  to  take  for  the  first  value 
of  «  a  number  equal  to  half  the  coefficient  of  x  with  its  sign 
changed.  Next,  values  of  x  differing  from  this  value  by  equal 
amo2ints  may  be  taken. 

Thus,  first  substituting  x  =  S,  it  is  found  that  j/  =  —  4,  Locating  the 
point  A  =  (3,  —  4).  Next  give  vakies  to  x  differing  from  i)  by  equal 
amounts,  as  2|  and  3|,  2  and  4,  1  and  5,  0  and  (5.     It  will  be  found  that 

for  x  =  4:  Rs  for  x  =  2, 
etc.  The  table  below  gives  a  record  of  the 
points  and  their  coordinates  : 


- 

"" 

, 

1 

-- 

E 

(  1? 

- 

t 

1 

\ 

il 

\ 

>n 

\ 

0 

d' 

Imp  N/ 

d' 

Q 

/ 

^r 

.^ 

^"fi 

] 

'1 

B  ' 

X 

y 

Points 

3 

-4 

A 

n,H 

-33 

B,B' 

2,  4 

-3 

C,  C 

1,  ^> 

0 

D,  D' 

0,6 

5 

E,E' 

326 


GRAPHIC  SOLUTIONS 


327 


Plotting  the  points  A;  B,  B' \   C\  C" ;  etc.,  whose  coordinates  are 
.en  in  the  preceding  table,  and  drawing  a  smooth  curve  through  them, 
obtain  the  graph  of  y  =  jc*-*  —  Oir  +  5  as  shown  in  the  figure. 


It  will  be  observed  that : 

When  x  =  3,  air— iJx-{-ij 

(jative  ordinate  PA. 

When  X  =  2   and   also   when   a;  =  4, 


—  4,  which  is  represented  by  the 


-  (i.f  -f-  />  =   -  8, 

which   is   represented    by  the   equal   negative   ordinates  MC 

and  NC. 

When  X  =  0  and  also  when  x  =  (>,  jf'  —  6  x*  -|-  5  =  5,  repre- 

iited  by  the  Q(\\xdl  poaitive  ordinates  OE  and  QE\ 

Thus,  it  is  seen  that  the  ordinates  change  sign  as  the  curve 

(losses  the  a^axis. 

At  D  and  at  D',  therefore,  where  the  ordinates  are  equal  to  0, 

t  lie  value  of  «^  —  6  ar  -|-  />  is  0,  and  the  abscissas  are  x  =  1  and 
=  o. 
Hence,  the  roots  of  the  given  equation  are  1  and  5. 

Note. — Half  the  coefficient  of  x  with  its  sign  changed,  the  number 
tirst  substituted  for  at,  is  half  the  sum  of  the  roots,  or  their  mean  value^ 
u  hen  the  coefficient  of  x'^  is  +  1.     This  will  be  shown  in  §  433. 

The  curve  obtained  by  i)lotting  the  graph  of  a^  —  6  ;c  -f  5,  or 
of  any  quadratic  expression  of  the  form  ax^  +bx  -{-c,  is  a 
parabola. 

419.   Let  it  be  required  to  solve  each  of  the  equations 
.r^_8a!  +  14=0,         (1) 
.r-8x-fl6=0,         (2) 
ar*- 8x4-18  =  0.         (3) 

The  graphs  corresponding  to 
(juations  (1),  (2),  and  (3), 
toiind  as  in  §  418,  are  marked 
1.  II,  and  III,  respectively. 

The  roots  of  (1).  are  seen  to  be 
o  F  =  2.6  and  O  W  =  5.4,  ap- 
I'loximately. 


~" 

I 

\ 

1 

'^\ 

1 

m 

I  •  " '  /3 

\       Tit 

\\\.;.//r 

u 

Kpnr 

\\ 

^iTVf 

\\ji7n- 

0 

v\ 

K    /W 

\^ 

J 

328 


GRAPHIC   SOLUTIONS 


Since  graph  II  has  only  one  point,  K,  in  common  with  the 
a^axis,  eqnation  (2)  appears  to  have 
only  one  root,  OK  —  4. 

But  it  will  be  observed  that  if 
graph  I,  which  represents  two  un- 
equal real  roots,  0  V  and  0  W,  were 
moved  upward  2  units,  it  would 
coincide  with  graph  II. 

During  this  process  the  unequal 
roots   of  (1),   OF  and   OW,  would 
approach  the  value  OK,  which  repre- 
sents the  roots  of  (2). 
Consequently,  the  roots  of  (2)  are  regarded  as  two  in  number. 
They  are  real  and  equal,  or  coincident. 

The  movement  of  the  graph  of  (1)  upward  the  distance  JK^  or  2  units, 
corresponds  to  completing  the  square  in  (1)  by  adding  2  to  each  member. 
Since  the  roots  of  the  resulting  equation,  x"^—  8x  +  16  =  2,  differ  from 
those  of  (2)  or  from  the  mean  value  OK  =  4,  by  ±  V2,  or  ±  y/JK,  it  is 
evident  that  the  roots  of  (1)  are  represented  graphically  by 


^II/ 

/ 

0 

v\ 

K 

V^ 

r 

J 

and 


0K+  yJJK^  4  +  V2  =  5.414+, 
0/1  -  \/J^=  4  -  V2  =  2.586— 


Since  graph  III  has  no  point  on  the  a;-axis,  there  are  no  real 
values  of  x  for  which  x-  —  8  a;  -f- 18  is  equal  to  zero ;  that  is,  (3) 
has  no  real  roots.     Consequently,  the  roots  are  imaginary. 

If  graph  III  were  moved  doicyivmrd  2  units,  it  would  coincide  with 
graph  II.  If  the  square  in  (3)  were  completed  by  suhtracting  2  from  each 
member,  the  roots  of  the  resulting  equation,  x"^  —  8x  -H  16  =  —  2,  would 
differ  from  the  mean  value  by  ±  V—  2,  or  i  y/ LK. 

Hence,  it  is  evident  that  the  roots  of  (3)  are  represented  graphically  by 


and 


0/t'+  VX/f  =4+  V-2, 
OK-  yTK^  4  -  V^^. 


The  points  J,  K,  and  L,  whose  ordinates  are  the  least  alge- 
braically that  any  points  in  the  respective  graphs  can  have, 
are  called  minimum  points. 


GRAPHIC   SOLUTIONS  329 

420.  When  the  coefficient  of  jt  is  -f  1,  it  is  evident  from  the 
preceding  discussion  that : 

Principles.  —  1.  Tfie  roots  of  a  quculratic  in  x  are  equal  to 
the  abscissa  of  the  minimum  ]}oint,  plus  or  minus  the  square  root 
of  the  ordinate  with  its  sign  changed. 

2.  If  the  minimum  point  lies  on  the  x-axis,  the  roots  are  real 
<uid  equal. 

'^.  If  the  minimum  point  lies  below  the  x-aj;is,  the  roots  are 
real  and  unequal. 

4.  If  the  minimum  point  lies  above  the  anixisj  the  roots  are 
imaginary. 

EXERCISES 

421     ^  ^      ^graphically  : 

1.  i  ,-1-3  =  0.  8.   ./•-  =  i\x-  10. 

2.  ar-6a;-f-7  =  0.  9.  ur^  +  4a;  +  2  =  0. 

3.  .r^  -  4  a;  =  -  2.  10.  ar  -f  3  x  +  4  =  0. 

4.  ^  =  2{x^l).  11.  ar^- 5  a: +  13  =  0. 

5.  a:^  -I-  2  (a-  -f  1)  =  0.  12.  .r-  -  2  a;  -f-  0  =  0. 

6.  a:2_4a;4-0  =  0.  13.  ar^-4x-l  =  0. 

7.  ar*  _  2  a;  -  2  =  0.  14.  ar^  +  7  a;  4-  14  =  0. 

15.  Solve   graphically  4a;— 2ar^-f  1  =0. 

Slcjc.kstion.  —  On  dividing  both  members  of  the  given  equation  by 
—  2,  the  coefficient  of  j-.  the  equation  becomes 

x-2  _  2  a:  -  i  =  0. 

The  roots  may  be  found  by  plotting  the  graph  of  y  =  x*^  —  2  x  -  J. 

Solve  graphically : 

16.  2x-2-h8a;  +  7  =  0.  18.    12  a;  -  4ar^  -  1  =  0. 

17.  2ar2-12a;-|-lo  =  0.  19.    11 -f  8  x  +  2  a;^  ^  0. 
NoTK.  —  Another  method  of  solving  quadratic  equations  graphically  is 


330 


GRAPHIC   SOLUTIONS 


422.    Graphs  of  quadratic  equations  in  x  and  /. 


EXERCISES 


1.   Construct  the  graph  of  the  equation  x^  -\-  y-  =  25. 


Solution.  —  Solving  for  y,    y  =  ±  v25  —  x^. 

Since  any  value  numerically  greater  than  5  substituted  for  x  will  make 
the  value  of  y  imaginary,  we  substitute  only  values  of  x  between  —  5  and 
4-  5.  The  corresponding  values  of  y,  or  of  ±  V'25  —  x^,  are  recorded  in  the 
table  below. 

It  will  be  observed  that  each  value  substituted  for  x,  except  ±  5,  gives 
two  values  of  y,  and  that  values  of  x  numerically  equal  give  the  same 
values  of  y ;  thus,  when  x  =  2,  y  =  ±  4.f),  and  also  when  x  =  —  2, 
y=±  4.6. 


X 

V 

0 

±  5 

±  1 

±4.9 

±  2 

±  4.6 

±  3 

±4 

±4 

±3 

±  5 

0 

1 

K*'^ 

^ 

^ 

■^ 

^ 

"V* 

/ 

IN 

^ 

/ 

\ 

\ 

' 

' 

\ 

J 

\ 

/ 

N 

/ 

s 

V 

^ 

y^ 

r^ 

I 

The  values  given  in  the  table  serve  to  locate  twenty  points  of  the  graph 
of  5c2  +  y^  =  25.  Plotting  these  points  and  drawing  a  smooth  curve 
through  them,  the  graph  is  apparently  a  circle.  It  may  be  proved  by 
geometry  that  this  graph  is  a  circle  whose  radius  is  5. 

The  graph  of  any  equation  of  the  form  jr-  +  /-  =  /^  is  a  circle 
whose  radius  is  r  and  whose  center  is  at  the  origin. 

2.  Construct  the  graph  of  the  equation  (x  —  2)-+  (y  —  3)-  =9. 

The  graph  of  any  equation  of  the  form  (jr  —  a)--\-  (/  —  by  =  r 
is  a  circle  whose  radius  is  r  and  whose  center  is  at  the 
point  (a,  h). 

3.  Construct  the  graph  of  the  equation  y-  =  3  a;  -f  9. 
Solution.  —  Solving  for  i/,  y  =  ±  V3  a;  +  9. 


GRAPHIC.  SOLUTIONS 


331 


It  will  be  ol)served  that  any  value  smaller  than  —  3  substituted  for  x 
will  make  y  iinagiuaiy  ;  consequently,  no  point  of  the  graph  lies  to  the 
left  ol  X  -  -  3.  Beginning  with  a;  =  —  3,  we  substitute  values  for  x  and 
Uelenniiie  the  corresponding  values  of  »/,  as  recorded  in  the  table  : 


X 

1     ' 

-8 

■          0 

-2 

±  l.V 

-1 

±2.4 

0 

±3 

1 

±3.6 

2 

±3.ft 

3 

±4.2 

— 

— 

— 

— 

— 

■~" 

^ 

..y^ 

r 

^ 

}j 

A 

n 

M 

V 

3 

Y 

' 

J 

'* 

1' 

> 

\ 

s 

k 

s 

k< 

'"V 

n 

•- 

Ploii,,^  points  and  drawin-;  a  smooth  curve  through  them,  the 

laph  obtaiue«l  is  apparently  a  parabola  (§418). 

The  graph  of  any  equation  of  the  form  /'  =  ax  +  c  is  a 
parabola. 

4.   Construct  the  graph  of  the  equation  9  «*  +  25  y*  =  225. 

Solution.  —  Solving  for  y,   y  =  ±  i  V226  —  9  x^. 

Since  any  value  numerically  greater  than  5  substituted  for  x  will  make 
the  value  of  y  imaginary,  no  point  of  the  graph  lies  farther  to  the  right 
or  to  the  left  of  the  origin  than  o  units  ;  consequently,  we  substitute  for  x 
(»nly  values  between  —  5  and  +  5, 

Corresponding  values  of  x  and  y  are  given  in  the  table : 


X 

y 

0 

±3 

±1 

±2.9 

±2 

±2.7 

±3 

±2.4 

±4 

±1.8 

±» 

0 

'  "T"  ■           '"         T " 

1 

**'*■?**# 

A  f^  ^    "^^-^  <,  ■^'^ 

2                ^\ 

%     -        ^^ 

it 

±   -. 

Plottinj;  these  twenty  points  and  drawing  a  smooth  curve  through  them, 
,  e  have  the  graph  of  9  x'^  +  25  y^  =  225,  which  is  called  an  ellipse. 


332 


GRAPHIC   SOLUTIONS 


The  graph  of  any  equation  of  the  form  b'-x"^  +  ary^  =  a^b"^  is  an 
ellipse. 

5.    Construct  the  graph  of  the  equation  4  £c-  —  9  2/^  =  36. 

Solution 

Solving  for  ?/,  y  =  ±^  V4  x^  —  36. 

Since  any  value  numerically  less  than  3  substituted  for  x  w^ill  make  the 
value  of  y  imaginary,  no  point  of  the  graph  lies  between  a;  =  +  3  and 
X  =  —  S  ;  consequently,  we  substitute  for  x  only  values  numerically  greater 
than  3. 

Corresponding  values  of  x  and  y  are  given  in  the  table : 


— 1 

1 

- 

X 

y 

> 

*4. 

/ 

<s. 

- 

>! 

K 

/ 

i^ 

±3 

±4 
±5 
±6 

0 
±  1.8 

±2.7 
±  3.5 

k 

s^ 

/ 

'<^ 

1 

1 

A 

\ 

1, 

4r^ 

S 

\ 

^ 

Ki 

\ 

K 

Plotting  these  fourteen  points,  it  is  found  that  half  of  them  are  on  one 
side  of  the  ?/-axis  and  half  on  the  other  side,  and  since  there  are  no  points 
of  the  curve  between  x  =  +  3  and  x  =  —  3,  the  graph  has  two  separate 
branches,  that  is,  it  is  discontinuous. 

Drawing  a  smooth  curve  through  each  gi'oup  of  points,  the  two  branches 
thus  constructed  constitute  the  graph  of  the  equation  4  x"^  —  9  y^  —  3(5^ 
which  is  an  hyperbola. 

The  graph  of  any  equation  of  the  form  b'^x^  —  aFy^  =  a-b^  is  an 
hyperbola.  An  hyperbola  has  two  branches  and  is  called  a 
discontinuous  curve. 

6.   Construct  the  graph  of  the  equation  xy  =  10. 

Solution 

Substituting  values  for  x  and  solving  for  y,  the  corresponding  values 
found  are  as  given  in  the  table  on  the  next  page. 


GRAPHIC   SOLUTIONS 


833 


X 

y 

z 

y 

1 

10 

-1 

-10 

2 

5 

-2 

-  o 

3 

8i 

-3 

-3J 

4 

2i 

-4 

-21 

'• 

2 

—  o 

-2 

0 

If 

-6 

-U 

7 

^ 

-7 

-If 

8 

u 

-8 

-u 

0 

H 

-9 

-u 

10 

1 

-10 

-1 

■ 

(I 

._J — I — 1—1)— I 1 — I — l— 


Plotting  these  points  and  drawing  a  smooth  curve  through  each  group 
•f  points,  the  two  branches  of  the  curve  found  constitute  the  graph  of  the 
•luation  xy  =  10,  which  is  an  hyperbola. 

The  graph  of  any  equation  of  the  form  jr^  =  c  is  an  hyperbola. 

Construct  the  graph  of  : 

10.    y.r^-l()^^  =  144. 


7.    x'  +  f  =  9. 

9.   9x'  +  16f  =  lU. 


11.  XV  =  12. 

12.  (.r-iy-'-h(y-2/=iC). 


423.  Summary. — The  types  of  equations  ;iii(l  their  respec- 
tive graphs,  here  summarized,  will  aid  the  student  in  plotting 
LTraj^hs,  but  he  will  meet  other  forms  of  equations  that  will 
liave  some  of  the  same  kinds  of  graplis,  tlie  varieties  in  equa- 
tions giving  rise  to  varieties  in  form,  size,  or  location  of  the 
graphs. 

For  example,  §  422,  exercises  1  and  2,  are  both  equations  of  the  circle, 
the  first  having  its  center  at  the  origin  and  the  second  at  the  point  (2,  3). 

It  is  possible  to  determine  many  characteristics  of  the  vari- 
«'us  graphs  from  their  equations  alone,  but  a  discussion  of 
this  is  beyond  the  province  of  algebra.  In  the  study  of 
1,'raphs,  therefore,  the  student  will  rely  principally  on  plotting 
;i  sufficient  number  of  points  to  determine  their  form  accurately. 


334 


GRAPHIC   SOLUTIONS 


The  following  types  have  been  studied: 
I.   ax  +  by=^c  (§  267) 

f  ajr^  +  6jr  +  c  =  0,  or  1 

y  =  ax-\-c 
b'x'-^ay  =  a'b' 


II. 
Ill 

IV 


V. 

VI. 

VII. 

VIII. 


b'x'^  —  ay-  =  a-b'- 


Straight  line 

Circle 

Circle 

Parabola 

Parabola 
Ellipse 
Hyperbola 
Hyperbola 


solution    of    simultaneous   equations   involving 


x/  =  c 

424.    Graphic 
quadratics. 

The  graphic  method  of  solving  simultaneous  equations  that 
involve  quadratics  is  precisely  the  same  as  for  simultaneous 
linear  equations  (§  271).  Construct  the  graph  of  each  equation, 
both  being  referred  to  the  same  axes,  and  determine  the  coordi- 
nates of  the  points  where  the  graphs  intersect.  If  they  do  not 
intersect,  interpret  this  fact. 

The  student  should  construct  the  following  graphs  for  him- 
self. Roots  are  expected  to  the  nearest  tenth  of  a  unit.  To 
obtain  this  degree  of  accuracy,  numerous  points  should  be 
plotted  and  a  scale  of  about  ^  inch  to  1  unit  should  be  used. 


425.     1.    Solve  graphically 


EXERCISES 

x'  +  f-  =  25, 

X  —  y  =  —  1. 

Solution.  —  Constructing  the  graphs 
of  these  equations,  we  find  the  first,  as  in 
§  422,  exercise  1,  to  be  a  circle  ;  and  the 
second,  as  in  §  267,  a  straight  fine. 

The  straight  line  intersects  the  circle 
in  two  points,  (—4,  —3)  and  (3,4). 
Hence,  there  are  two  solutions, 

X  =  —4,  ?/  —  —  3  ;  and  ic  =  3,  ?/  =  4. 

Test.  —  The  student  may  test  the 
roots  found  graphically  by  performing 
the  numerical  solution. 


1   i 

■ 

1 

^— 

1 

/ 

, 

/ 

.*i^ 

k, 

/ 

nX" 

y 

/ 

\ 

V 

/ 

\ 

/ 

/ 

\ 

/ 

;< 

/ 

\ 

~^ 

/ 

\ 

/ 

^ 

y 

/ 

^ 

-^ 

y 

- 

1 

1 

1 

P..J 

GRAPHIC   SOLUTIONS 


335 


2 


;rai)lii<'all; 


c; 


9ar--h-'''>y'  =  226, 


St)Li  TioN. — On    constructing     the 

iplis  (for  the  first,  see  exercise  4, 
§422),  it  is  found  that  they  intersect 
at  the  points  x  =  •).!.  y=  2,  x.  =  —  3.7, 
y  =  2. 

Since  the  graphs  liave  these  two 
points  in  common,  and  no  others,  their 
■   "trdinates  are  the  only  values  of  x  and 

I  hat  satisfy  both  equations,  and  are 
MU'  roots  sou<;ht. 

<  )b8erve  that  the  pairs  of  values  z  = 
7.   y  =  2and  x  =  —  3.7,   y  =  2,   are 

d,  and  different,  or  unequal. 

XoTE. — The  roots  are  estimated  to  the  nearest  tenth  ;  their  accuracy 
limy  be  tested  by  performing  the  numerical  solution. 

9a-*  4- 25/=  225, 


—[ 

xrS- 

4-+-^-— "fff- 

tS-""--^" 

__j: __: 

— t —      ------ 

^^           ^i 

^-i-rr^^' 

wP^-^ 

1 

1 

1 

3.    Solve  j^raphioally 


V  =  3. 


SoLTTioN.  —  Imagine  the  straight  line  y  —2  in  the  figure  for  exercise 
-'  to  move  upward  until  it  coincides  with  the  line  y  =  3,    The  real  unequal 

•ts  represented  by  the  coordinates  of  the  points  of  intersection  approach 
luality,  and  when  the  line  liecomes  the  tangent  line  y  =  3,  they  coincide. 
IltMice,  the  given  system  of  equations  has  ttoo  real  equal  roots,  x  =  0, 
/  =  :{,  and  x  =  0,  y  =  3. 


4.    Find  the  nature  of  the  roots  of 


.'  _  9*> 


SonTioN.  —  Imagine  the  straight  line  y  =  2  in  the  figure  for  exercise 
:<»  move  upward  until  it  coincides  with  the  line  y  =  4.     The  graphs  will 

IS.'  to  have  any  points  in  common,  showing  that  the  given  equations 
li;ivi   no  common  real  values  of  x  and  y. 

It  is  shown  by  the  numerical  solution  of  the  equations  that  there  are 
t  wo  roots  and  that  both  are  imaginary. 

A  system  of  two  independent  simultaneous  equations  in  x  and 
'/,  one  simple  and  the  other  quadratic^  has  two  roots. 

TJie  roots  are  real  and  unequal  if  the  graphs  intersect,  real  and 
"/nal  if  the  graphs  are  tangent  to  each  other,  and  imxiginary  if  the 
iji'dplis  have  no  points  in  commjon. 


336 


GRAPHIC   SOLUTIONS 


-r 

n 

"1 

|,o. 

A>r 

^ 

S 

1 

V, 

N 

*/ 

.. 

0 

\ 

^ 

k-V 

- 

^ 

r 

~v 

V 

U 

- 

\t'^ 

"'A 

r 

J 

\ 

A\ 

/ 

y 

/ 

^' 

•v 

• 

- 

A 

y^ 

\ 

^ 

^ 

^ 

^ 

1 

5.  Solve  graphically 

Solution.  —  The  graphs  (see  ex- 
ercises 5  and  1,  §  422)  show  that 
both  of  the  given  equations  are  satis- 
fied by  four  different  pairs  of  real 
values  of  x  and  y  : 

JtK  =  4.5;       4.5;  -4.5;  -4.5; 

ly  =  2.2;  -2.2;  -2.2;       2.2. 

6.  What  would  be  the  nature  of  the  roots  in  exercise  5,  if 
the  second  equation  were  ar  -f  ?/2  =  9  ? 

A  system  of  tivo  independent  simultaneous  quadratic  equations 
in  X  and  y  has  four  roots. 

An  intersection  of  the  grcqihs  represents  a  real  root,  and  a 
j)oint  of  tangeyicy,  a  pair  of  equal  real  roots.  If  there  are  less 
than  four  real  roots,  the  other  roots  are  imaginary. 

Find  by  graphic  methods,  to  the  nearest  tenth,  the  real  roots 
of  the  following,  and  the  number  of  imaginary  roots,  if  there 
are  any.     Discuss  the  graphs  and  the  roots. 


7. 


8. 


10. 


11.      ^■ 


4a^-92/'  =  36, 
x-^y  =  l. 

.4a.'2  +  9/  =  36. 
9.r^-fl6/  =  144, 

'ar-h/  =  4, 
.  .r  =  2/  -  5. 
ra^-4?/2  =  4, 


12. 


[  ^2  _j_  ^-  _  4^ 

i»  -  2/  =  2, 
xy  =  -l. 


13. 


14. 


15. 


16. 


17. 


18. 


r4a;2_9^2^30^ 
I  4  2/  =  .r-  -  16. 
f9x2-f-ir)?/2  =  144, 
l3aT-f-47/  =  12. 

r.x-^-h/  =  9, 

1 2/  =  or  —  o  X  +  6- 

U-  =  ?/-  -h  5  ?/  +  6. 

y  =  x^-4., 

^  =  (^  +  l)(2/  +  4). 
f  y  =ar  —  5  a;  +  4, 
lx=/-4i/  +  3. 


GRAPHIC   SOLUTIONS  337 

19      fy'  +  x-^  +  ?/-2^4-l=0, 

It  is  nut  possible  to  solve  any  two  simultaneous  equations 
in  X  and  y,  that  involve  quadratics,  by  quadratic  methods, 
but  approximate  values  of  the  real  roots  may  always  be  found 
hy  the  graphic  method. 

Solve  the  following  by  both  methods,  if  you  can  : 
20.     {->.^  =  l^.^  21.      ^^  +  ^='' 


0:^^  +  ^  =  26. 


[/  +  a:=ll. 


426.  Another  graphic  method  of  solving  quadratic  equations 
in  jr  (§  418). 

It  Inis  been  seen  that  the  real  roots  of  simidtaneous  equor 
(ions  are  the  coordinates  of  the  points  where  their  graphs  inter- 
sect or  are  tangent  to  each  other,  and  that  when  there  is  no 
point  in  common,  the  roots  are  imaginary. 

In  §§  418-421,  it  was  found  that  the  real  roots  of  a  quadratic 
"juation  were  the  abscissas  of  the  points  where  the  graph  of 
the  quadratic  expression  crossed  or  touched  the  ar-axis,  and 
tliat  when  it  had  no  point  in  common  with  the  aj-axis,  the  roots 
were  imaginary. 

In  other  words  the  solution  of  a  quadratic  equation  in  x  was 
made  to  depend  upon  the  solution  of  the  simultaneous  system, 

f  .V  =  «^  -\-bx-{-c,  (a  parabola) 

\y  =  0,  (a  straight  line) 

the  second  being  the  equation  of  the  ovaxis. 

In  the  following  method,  by  substituting  y  for  x^  in  the  given 
-quation, 

ax^  +  bx-\-c  =  0, 

the  equation  is  divided  into  the  simultaneous  system, 

,y  f  ay  -f  6ic  -f  c  =  0,  (a  straight  line) 

\y  =  x^.  (a  parabola) 

The  solution  of  this  system  for  x  gives  the  required  roots  of 


338 


GRAPHIC   SOLUTIONS 


It  will  be  observed  that  whether  system  I  or  II  is  used, 
the  solution  requires  the  construction  of  a  parabola  and  a 
straight  line,  but  the  advantage  of  using  II  instead  of  I  lies  in 
the  fact  that  the  parabola  y  =  x-  is  the  same  for  all  quadratic 
equations  in  x  and  when  once  constructed  can  be  used  for  solv- 
ing any  number  of  equations,  while  with  I  a  different  parabola 
must  be  constructed  for  each  equation  solved. 


EXERCISES 

427.    1.  Solve  graphically  the  equation  x^  —  2x  —  S  =  0. 
Solution 

Substituting  y  for  ic^,  we  have 

2/-2a;-8=0. 
Consequently,  the  values  of  x 
that  satisfy  the  system, 

?/-2x-8=0, 

are  the  same  as  those  that  satisfy 
the  given  equation. 

Constructing  the  graph  of 
y  =  aj2,  we  have  the  parabola 
shown  in  the  figure. 

Constructing  the  graph  of 
?/  —  2x  —  8  =  0,  a  straight  line,  we 
find  that  it  intersects  the  parabola 
at  X  =  —  2  and  x  =  4:. 

Hence,  the  roots  of  the  equation 
x2  _  2  X  -  8  =  0  are  -  2  and  4. 

Solve  graphically,  giving  roots  to  the  nearest  tenth : 


T^ 

4                   ^ 

^                   -U 

l- 

i 

i-           .4 

I          ^54 

I         J/I 

4     "^7  i 

^  t 

7    4^ 

V-       f     t 

V    1-    ^ 

\    t       t 

42         ^ 

H 

A-      4^ 

J         tx    t 

7  \   f 

jL      ^2 

7             -T 

rr                   \ 

IL 

2.  x^-^x-2  =  0. 

3.  x^~x-6  =  0. 

4.  x'-3x-4.  =  0. 

5.  a;2-2i«-15  =  0. 

6.  a:-  +  5ic  — 14  =  0. 

7.  a^_7ar-18  =  0. 


8.  2x'-x  =  6. 

9.  2x'-x-W  =  0. 

10.  3a^  +  5a;-28  =  0. 

11.  6x'-7x-20  =  0. 

12.  8a^ 4- 14.^-15  =  0. 

13.  15a;2  +  2x-20  =  0. 


PROPERTIES   OF   QUADRATIC   EQUATIONS 


428.  Nature  of  the  roots. 

Ill  the  following  discussion  the  student  should  keep  in 
mind  the  distinctions  between  rational  and  irrational,  real  i^nd 
imaginary. 

For  example,  2  and  VT  are  rational  and  also  real ;  V2  and  VS  are 
irrational,  but  real;  V—  2  and  V—  5  are  irrational  and  also  imayinary. 

429.  Every  quadratic  equation  may  be  reduced  to  the  form 

aa?  -f  bx  +  0  =  0, 

in  which  a  is  positive  and  h  and  c  are  positive  or  negative. 
Denote  the  roots  by  r^  and  r^     Then,  §  .390, 


and  r^ 


2a  "^  2a 

An  examination  of  the  above  values  of  r^  and  r^  will  show 
that  the  nature  of  the  roots,  as  real  or  imaginary,  rational  or 
irrational,  may  be  determined  by  observing  whether  V6^  —  4«c 
is  real  or  imaginary,  rational  or  irrational.     Hence, 

i  * i;  I  \  <  I PLE8.  —  In  any  quadratic  equation,  aa^  -\-  bx  -^  c  =  0, 
when  o,  6,  and  c  represent  real  and  rational  numbers : 

1.  7/*6*  —  4  ac  is  positive,  the  roots  are  real  and  unequal. 

2.  Ifb^  —  4ac  equals  zero,  the  roots  are  real  and  equal. 

3.  Ifb'  —  A  ac  is  negative,  the  roots  are  imaginary. 

4.  Ifb^—iac  is  a  perfect  square  or  equals  zero,  tJie  roots  are 
rational;  otherwise,  they  arc  irrational. 

430.  The  expression  6^  —  4  ac  is  called  the  discriminant  of 
the  quadratic  equation  a^ -\'  bx-\-  c  =  0. 


340         PROPERTIES   OF   QUADRATIC   EQUATIONS 

431.  If  a  is  positive  and  b  and  c  are  positive  or  negative,  the 
signs  of  the  roots  of  ajr  -|-  6a?  4-  c  =  0,  that  is,  the  signs  of 

-  6  4-  -y/b'  -4ac       ,           -  6  -  Vb^  -4ac 
n  = 2  «  ^"-  ^^^^  ^'-^  = 2-a ' 

may  be  determined  from  the  signs  of  b  and  c. 

Thus,  if  c  is  positive,  —  6  is  numerically  greater  than 
±  V&'  —  4  ac,  whence  both  roots  have  the  sign  of  —  6 ;  if  c  is 
negative,  —  6  is  numerically  less  than  ±  V6^  —  4  ac,  whence 
i\  is  positive  and  r^  is  negative.  The  root  having  the  sign 
opposite  to  that  of  b  is  the  greater  numerically.     Hence, 

Prixciple.  —  If  c  is  2^ositive,  both  roots  have  the  sign  opposite 
to  that  ofb;  ifc  is  negative,  the  roots  have  opposite  signs,  and  the 
numerically  greater  root  has  the  sign  opposite  to  that  of  b. 

Note.  — If  6  =0,  the  roots  have  opposite  signs.     (See  also  §  378.) 
EXERCISES 

432.  1.    What  is  the  nature  of  the  roots  of  a;^  —  7  x-  —  8  =  0? 

Solution.  —  Since  Ij^  —  'iac  —  49  +  32  =:  81  =  9'^^  a  positive  number  and 
a  perfect  square,  by  §  429,  Prin.  1,  the  roots  are  real  and  unequal ;  and  by 
Prin.  4,  rational. 

Since  c  is  negative,  by  §  431,  Prin.,  the  roots  have  opposite  signs  and,  h 
being  negative,  the  positive  root  is  the  greater  numerically. 

2.  What  is  the  nature  of  the  roots  of  3  .^'-  +  5  ic  +  3  =  0  ? 

Solution-.  —Since  &2  _  4  q^c  ==  25  -  36  =  —  11,  a  negative  number,  by 
§  429,  Prin,  3,  both  roots  are  imaginary. 

Find,  without  solving,  the  nature  of  the  roots  of : 

3.  a.-2- 5a.- -75:^0.  8.    A.x'  -  4.x +  1  =  0. 

4.  ^  +  5a;_f_6  =  0.  9.    4a!2_^6a;-4  =  0. 

5.  ^^1  x-^(}  =  0.  10.    X-  +  x-\-2  =  0. 

Q,    x^-^x^n  =  0.  l\.    4.x-  +  lQ>x  +  l  =  0. 

7.    X-  +  3  a;  —  5  =  0.  12.    9  a;-  +  12  a;  +  4  =  0. 


PROPERTIES   OF   QUADRATIC   EQUATIONS  11 

13.  For  what  values  of  m  will  the  equation 

2x^  +  S7nx-{-2=:0 
have  equal  roots  ?  imaginary  roots  ? 

Solution 

The  roots  will  be  equal,  if  the  discriminant  equals  zero  (§  420.  Prin.  2^ ; 
t hat  is,  if  (:J  //«)■■*  -4-2.2  =  0, 

'  <r,  solving,  if  m  =  f  or  —  J. 

The  roots  will  be  imaginary,  if  the  discriminant  is  negative  (§  429, 
Prin.  3)  ;  that  is,  if  (3  m)'^  —  4  •  2  •  2  is  negative, 

which  will  be  true  when  m  is  numerically  less  than  |. 

14.  For  what  values  of  m  will  9  a^  —  omx  -\-  25  =  0  have 
|ual  roots?  real  roots?  imaginary  roots? 

15.  For  wh'dt  values  of  a  will  the  roots  of  the  equation 

4ar^_2(a-3)a:-}-l  =  0 
Ke  real  and  equal?  real  and  unequal?  imaginary? 

16.  Find  the  values  of  wi  for  which  the  roots  of  the  equation 

4  y-  +  mx  4-  .r  -i-  1  =  0 
are  equal.     What  are  tlie  corresponding  values  of  «? 

17.  For  what  values  of  n  are  the  roots  of  the  equation 

:5  ar^  -H  1  =  7i(4  x  —  2  ar*  —  1)  real  and  equal  ? 

18.  For  what  value  of  a  are  the  roots  of  the  equation 

ax"-  (a-l)x  +  l=0 
numerically  equal  but  opposite  in  sign?     Find  the  roots  for 
this  value  of  a. 

19.  For   what  value  of  d  has  ar^  -f  (2  —  (l)x  =  3d*  —  27  a 
ro  root?     Find  both  roots  for  this  value  of  d. 

20.  For  what  values  of  m  will  the  roots  of  the  equation 

(m  +  f)«*  -  2(m  +  1)0!  -f  2  =  0  be  equal? 

21.  Solve  the  simultaneous  equations  for  x  and  y 

r3a*-4/  =  8, 
\5(x-k)  -4y  =  0. 
For  what  values  of  k  are  the  roots  real  ?  imaginary  ?  equal  ? 


342         niOPERTIES   OF   QUADRATIC   EQUATIONS 

433.    Relation  of  roots  and  coefficients. 

Any  quadratic  equation,  as  aay^  -\- bx  -\-  c  =  0,  may  be  reduced, 
by  dividing  both,  members  by  the  coefficient  of  a^,  to  the  form 
x^  4-  2^^  +  ^  =  ^>  whose  roots  by  actual  solution-  are  found  to  be 


_-p-[.  ^p'^-^.q       .      _  -p-  ^p'--4:q 
1 1  — — ana  ?  2  — ^ 

Adding  the  roots,     r^  -\-  r^  =  — ^—^  =  —  P- 

Multiplying  the  roots,  r^r^  =  ^  ~~      *~ —     —  ^• 

Hence,  we  have  the  following : 

Principle.  —  The  sum  of  the  roots  of  a  quadratic  equation 
having  the  form  x^  -\-  px  -\-  q  =  0  is  equal  to  the  coefficient  of  x 
with  its  sign  changed,  and  their  product  is  equal  to  the  absolute 
term: 

434.  Formation  of  quadratic  equations. 

Substituting  —  (ri  +  r^)  for  p,  and  r^r^  for  q  (§  433)  in  the 
equation  x^  -\-  px  +  q  =  0,  we  have 

^'^  —  (^'1  +  ^'2)^  +  ni\  =  0. 
Expanding,  x^  —  r^x—  r^  -\-  r^r^  =  0. 

Factoring,  (x  —  r^{x  —  r^  =  0. 

Hence,  to  form  a  quadratic  equation  whose  roots  are  given : 

Subtract  each  root  from  x  and  place  the  pi'oduct  of  the  remain- 
ders equal  to  zero. 

EXERCISES 

435.  1.    Form  an  equation  whose  roots  are  —5  and  2. 
Solution,     (x  +  5)  (x  -  2)  =  0,  or  x^  -f  .3  x  -  10  =  0. 

Or,  since  the  sum  of  the  roots  with  their  signs  changed  is  +  5  —  2, 
or  3,  and  the  product  of  the  roots  is  -  10,  (§  433)  the  equation  is 
x2  +  3  X  -  10  =  0. 


4. 

3,  -i. 

10. 

h-^\,b-\. 

5. 

hi- 

11. 

a  -f  6,  a  —  6. 

6. 

-2,  -». 

12. 

Va  -  V6,  V6. 

7. 

-  *)  -  -y- 

13. 

H«±  V6). 

PROPERTIES  OF   QUADRATIC   EQUATIONS        343 

Form  the  equation  whose  roots  are : 

2.  (),  4.  8.   rt,  -3  a.  14.   3-fV2,  3-V2. 

3.  T),  -.S.  9.   tt-|-2,a-2.  15.    2-V5,2+V5. 

16.  2±V3. 

17.  -.V(3±V6). 

18.  \(-l±V2), 

19.  a(2±2V5). 

20.  What  is  the  sum  of  the  roots  of  2  7n^ji^  —  (5  m  —  1) a;=  6 ? 
I'or  what  values  of  m  is  the  sum  equal  to  2? 

21.  When  one  of  the  roots  of  ax^  +  bx-\-c  =  0  is  twice  the 
other,  what  is  the  relation  of  b^  to  a  and  c? 

Solution 

Writing  ax'^  -f-  6x  +  c  =  0  in  the  form 

x2  +  ^x  +  ^  =  0,  (1) 

a        a 

and  representing  the  roots  by  r  and  2  r,  we  have 

r  +  2r  =  3r  =  --,  (2) 

a 

and  r  •  2  r  =  2  r'^  =  '^ .  (3) 

a 

On  sub.siituting  the  vahie  of  r  obtained  from  (2)  in  (3)  and  reducing, 

62=1  ar. 

22.  Obtain  an  equation  expressing  the  condition  that  one 
K  »ot  of  4  a:^  —  3  ox  4-  6  =  3  is  twice  the  other. 

23.  Find  the  condition  that  one  root  of  ax^  +  bx  -f  c  =  0 
sliall  be  greater  than  the  other  by  3. 

24.  When  one  root  of  the  general  quadratic  cu? -{-bx-\-c  =  0 
is  the  reciprocal  of  the  other,  what  is  tlie  relation  between  a 
and  c? 

25.  If    the   roots  of   aj? -\-bx-^  c=  0  are  r,  and  rg,  write  an 
1  nation  whose  roots  are  —  r^  and  —  r.^. 


344      propj:rties  of  quadratic  equations 

23.  Obtain  the  sum  of  the  squares  of  the  roots  of 
2  a;^  — 12  ic  +  3  =  0,  without  solving  the  equation. 

Solution 

Sum  of  roots  =  ri  -\-  ro  =  6.  (1) 

Product  of  roots  ■=  viv^  =  f-  (2) 

Squaring  (1),  n^  +  r-i^  +  2  nr^  =  36.  (3) 

(2)  X  2,  2  nro  =  3.  (4) 

(3) -(4),  n^+r^^^SS. 

Find,  without  solving  the  equation: 

27.  The  sum  of  the  squares  of  the  roots  of  or —  5  x— 6  =  0. 

28.  The  sum  of  the  cubes  of  the  roots  of  2  a;^  —  3  .^'  + 1  =  0. 

29.  The  difference  between  the  roots  of  12  a^  +  a;  —  1  =  0. 

30.  The  square  root  of  the  sum  of  the  squares  of  the  roots 
of  a.-2-7aj  +  12  =  0. 

31.  The  sum  of  the  reciprocals  of  the  roots  of  ax--}-bx-\-c=0. 

Suggestion-.  -  +  -  =  ^X+J!? . 

ri      To        rir2 

32.  The  difference  between  the  reciprocals  of  the  roots  of 
8  a^-  -  10  a;  +  3  =  0. 

436.    The  number  of  roots  of  a  quadratic  equation. 

It  has  been  seen  (§  433)  that  any  quadratic  equation  may 
be  reduced  to  the  form  x~  -f  pa?  -{-  q  =  0,  which  has  ttco  roots, 
as  Ti  and  j'o.  To  show  that  the  equation  cannot  have  more 
than  two  roots,  write  it  in  the  form  given  in  §  434,  namely, 

(x-r,)(x-r,)  =  0.  (1) 

If  the  equation  has  a  third  root,  suppose  it  is  r^. 
Substituting  r^  for  x  in  (1),  we  have 
(7'3-ri)03-r2)=0, 
which  is  impossible,  if  r  differs  from  both  Vi  and  rg.      Hence, 

Prixciple.  —  A  quadratic  equation  has  tivo  and  only  two 
roots, 


PROPERTIES  OF   QUADRATIC   EQUATIONS        345 

437.   Factoring  by  completing  the  square. 

The  method  of  factoring  is  useful  in  solving  quadratic  equa- 
t  ions  when  the  factors  are  rational  and  readily  seen.  In  more 
(lirticult  cases  we  complete  the  square.  This  more  powerful 
method  is  useful  also  in  factoring  quadratic  expressions  the 
factors  of  which  are  irrational  or  otherwise  difficult  to  obtain. 


EXERCISES 

438.    1.    Factor  2xr  -\-  5x-3. 

Solution 
Let  2a;-*  + 5x-3  =  0. 

Dividing  by  2,  etc.,  a;*  +  fa;  =  |. 

Completing  the  square,        x'^  +  ^^  +  H^  H- 
Solving,  a;  =  i  or  —  3. 

Forming  an  e(jiiatioii  having  these  roots,  §  434, 

(x-i)(x  +  3)  =0. 
Multiplying  by  2  because  we  divided  by  2, 

(2x-  l)(x  +  3)  =  2x-''  +  6x-3  =  0. 
Hence,  the  factors  of2x2  +  6x  —  3are2x— 1  and  x  +  3. 


Factor : 

2.   ia^-4x-S. 

6. 

7a^  +  rSx-2. 

3.    5  ar'  -1-  3  X  -  2. 

7. 

24ar^-10.c-25. 

4.   3a^  +  14a;-5. 

8. 

10x^  +  21  a;-  10. 

5.    8jr'-14a-  +  3. 

9. 

15ar^-5.5x-l. 

10.    Factor  ar  H- 2  a; - 

-4. 

Solution 
Let  x2  +  2  X  -  4  =  0. 

Completing  the  square,  x^  +  2  x  +  1  =  6. 

Solving,  X  =  —  1  +  V6  or  -  1  -  V 5. 

Hence,  §  434,  (x  +  1  -  \/5)(  x  +  1  +  y/l)  =  x2  +  2x-4=0. 
That  is.  the  factors  of  x-^  +  2  x  -  4  are  x  +  1  -  -/S  and  x  -f  1  +  V5, 


346        PROPERTIES   OF   QUADRATIC   EQUATIONS 

Factor : 

11.  .^  +  4aj-6.  14.  x^  +  x-{-l. 

12.  y'-6y +  3.  15.  a^-\-3a—5. 

13.  z^—5z  —  l.  16.  ^^  +  3^  +  7. 

17.  Factor  2-'Sx-2a^. 

Suggestion.  —  Since  2  —  ^x  —  2x^  =  — 2(x2  +  |x  —  1),  factor  x^  +  |  x 
—  1,  in  which  the  coefficient  of  x^  is  +  1,  and  multiply  the  result  by  —  2. 

18.  2i«2_^2x-l.  21.    9a2-12a  +  5. 

19.  9x^  —  4:X-^l.  22.    16^(1  — v)— 9. 

20.  24a^-16.T--3.  23.    16(3  +  n)  +  3  ril 

24.  Factor  100  a.-^  +  70  ojt/ - 119 /. 

Suggestion.  —  The  coefficient  of  x"2  being  a  perfect  square,  complete  the 
square  directly  ;  do  not  divide  by  100. 

25.  4  62  _  48  5 +  143.  28.   16p(p  +  l)  -  1517. 

26.  9  ?'2  _  12  r  +  437.  29.    25  e^- 2  h(o  e  -  2  h). 

27.  4a2  4_i2a-135.  30.    Sh(4:k-3  h) -7  JcK 

31.    Factor  o;^ -h  4  ic^  +  8  a;2  +  8  ;c  —  5. 

Solution 

Let  x*  +  4  x3  +  8  x2  +  8  X  -  5  =  0. 

Completing  the  square, 

(x4  +  4  x^  +  4  x2)  +  4(x2  +  2  X)  +  4  =  9. 
Extracting  the  square  root,  x^  +  2  x  +  2  =  3  or  —  3. 

.-.  x4  +  4  x3  +  8  x2  +  8  X  -  6  =  (x2  +  2  X  +  2  -  3)  (X--2  +  2  X  +  2  +  3) 
:=  (x2  +  2  X  -  1)  (x-'^  +  2  X  +  5). 

Factor  the  following  polynomials  : 

32.  x'-\-6:i^  +  llx^-\-6x-S. 

33.  x^-^2x'  +  5x'-\-Sx^  +  Sx''-{-Sx  +  3. 

34.  x^-4.a^  +  6x'-\-6o^-ldx-  +  10x  +  9. 

35.  4  ic«  4- 12  a^  +  25  x'  +  40  a^=^  +  40  x"^  +  32  x  +  15. 


PKOPERTIES  OF  QUADRATIC   EQUATIONS         347 

36.  Resolve  x^  -h  1  into  factors  of  the  second  degree. 

Solution 
a^+ l=x*  + 2x2  +  1 -2a;2 
=  (x2  +  1)2  _  (x  v/2)2 
=  (a;2  +  x  V2  +  1)(  a;2  -  a;  Vl  +  1). 

XoTE.  —  Each  of  these  quadratic  factors  may  be  resolved  into  two 
tors  of  the  first  degree  by  completing  the  square.     The  factors  are: 

,^  +  ^V2  +  iV':r2),     (a;  +  i  >/2  -  J  V^:2),     (X  -  1  V 2  +  i  V^^r2), 

and  (a;  -  J  V2  -  i  \^32). 

Resolve  into  quadratic  factors: 

37.  x*  +  16.  39.   a;^  +  2aV  +  4a*. 

38.  r/^H-6*.  40.    V*  —  4nv—2n\ 

439.  Values  of  a  quadratic  expression. 

An  expression  that  has  different  values  corresponding  to 
'  ■  tferent  values  of  x  is  called  a  function  of  x. 

-  2x  is  a  function  of  x,  for  when  a;  =  1,  2,  3,  ••-,  a;^— 2x  =  —  1,  0,  3,  •••. 

In  the  following  discussions  only  reed  values  of  x  are  con- 
sidered. 

BXBRCISBS 

440.  1.    What   values  has  the  function  a^  — 2  a;  — 3  corre- 
sponding to  very  large  positive  or  negative  values  of  a;? 

Discission.  —  When  x  is  very  large  and  either  positive  or  negative, 
value  of  x*  —  2  X  —  3  is  approximately  equal  to  that  of  its  largest  term, 
Thus  when  x  =  ±  100,  x*  —  2x  —  3  =  10,000,  approximately  ;  when 
x=  ±  1000,  x2  -  2x  -  3  =  1,000,000,  approximately. 

Since  x*  is  always  positive,  whether  x  is  positive  or  negative,  for  very 
^e  values  of  X,  x"^  —  2  x  —  3  is  very  large,  and  positive  ;  and  by  making 
iriTP  enough  we  can  make  x*  —  2x  -  3  greater  than  any  number  that 
miy  ir  as  lulled,  however  great. 

A  number  that  may  become  greater  than  any  assignable 
1  lumber  is  called  an  infinite  number. 

The  symbol  of  an  infinite  number  is  oc  ,  read  '  infinity.^ 


348        PROPERTIES   OF   QUADRATIC   EQUATIONS 


2.    Interpret  the  conclusion  of  exercise  1  graphically. 


■■-] 

iiijiiiiiifii: 

--------T--- 

====F==i==== 

\—Wl — 

_      j-_  ±±  : 

=====^:=^====: 

iiiii=giiiii 

Discussion.  —  Draw  the  graph  of  x^ 
—  2  X  —  3,  plotting  values  of  x  as  abscis- 
sas and  values  of  x^  —  2  x  —  3  as  ordi- 
nates  (§418). 

In  the  discussion  of  exercise  1,  it  is  seen 
that  when  x  =  —  c»,  and  also  when 
X  =  +  CO ,  x'^— 2  X  —  3  =  +  00  ;  that  is,  when 
X  increases  without  limit,  either  in  the 
negative  direction,  or  in  the  positive  direc- 
tion along  the  x-axis,  x^  —  2  x  —  3,  repre- 
sented by  ordinates  to  the  curve,  increases 
without  limit  in  the  positive  direction. 


Referring  to  the  graph  of  x^  —  2  x  —  ^  and  observing  the 
form  of  the  function  itself,  a  brief  discussion  for  real  values 
of  X  may  be  given  as  follows : 

(a)  As  X  increases  continuously  from  —  oo  to  4-1,  x^  —  2x—Z 
decreases  continuously  from  -|-  oo  to  its  minimum  value,  —  4, 
crossing  the  a;-axis  at  ic  —  —  1,  vi^hich  is  therefore  a  root  of  the 
equation  x^  —  2  ic  —  3  =  0. 

(Ij)  As  X  increases  continuously  from  -f  1  to  +  oo  ,  a;^  —  2  ic— 3 
increases  continuously  from  its  minimum  value,  —  4,  to  -|-  oo , 
crossing  the  i«-axis  at  x  —  3,  which  is  therefore  the  other  root 
of  the  equation  cc-  —  2  ic  —  3  =  0. 

(c)  The  function  is  positive  for  all  values  of  x  outside  the 
limits  x  =  —  \  and  a;  =  3,  and  negative  for  all  values  of  x 
within  these  limits. 

When  the  coefficient  of  y?-  is  +  1,  the  abscissa  of  the  minimum  point  is 
half  the  coefficient  of  x  with  its  sign  changed  (§  418). 

In  a  similar  way  discuss  the  following  functions : 

3.  a:2-5x-h6.  6.    ar^-hSa;-}-^ 

4.  0.-2- 2  X -8.  7.    a^-9. 

5.  a^-f2a;  — 15.  %.    q^-\-x-\-\. 


PROPERTIES  OF   QUADRATIC    EQUATIONS         349 

9.    Find  the  maximuin  value  of  3  -\-'2  x  —  x-. 


'm 


First  Solution 
Since 8  +  2x-x2=_ (a;*i- 2  x  -3),  and 
X'  —  2  X  —  3  has  a  minimum  value  at  x  =  1 
(exercise  2),  the    given    function  has  a 
maximum  value  at  x  =  1. 

When  X  =  1,  3  +  2  x  —  x-^  =  4,  the  wox- 
i  1,1  ion  value. 

Second  Solution 

Let  3  +  2  X  -  x*  =  y. 
Solving  for  X,  X  =  1  ±  V4  —  y. 
Since  x  must  be  real,  4  —  y  =  0  or  a 
positive  number. 
If  4  -  y  =  0,  y  =  4. 

If  4  —  y  =  a  positive  numl^er,  y  is  hss  than  4. 
Therefore,  4  is  the  maximum  value  of  the  function. 

10.  Complete  the  discussion  of  the  values  of  3  4-  2  a;  —  ic^. 
Discuss  the  values  of  the  following : 

11.  x-\-(S-x'.  14.   2^^nx-Z. 

12.  o-A.x-x'.  15.   2a:^-h3a;-f 2. 

13.  4ar'-lGa;4-lo.  16.   4.i--6-x*. 

17.  For  what  values  of  r  is  ./•-  —  .")  x  +  0  positive? 

S<.»Ll   1  H»S 

a;2  _  5  X  +  6  =  (x  -  2) {j-  -  .-i). 
, .!  —  5  a;  +  6  is  i>ositive  when  both  factors  are  jwaitive  or  when  both  are 
-iitive;  that  is,  when  x  is  less  than  2  or  greater  than  :i  these  values 
iig  the  roots  of  the  equation  x*  -  ox  +  0  =  0. 

18.  For  what  values  of  x  is  jr  — .Su-  — 28  positive?  nega- 
tive? 

19.  Show  that  ic*  —  6  x  -f  12  is  positive  for  all  real  values  of 
r .     What  is  the  nature  of  the  roots  ofa^-6a;-l-12  =  0? 

20.  Show  that  x  —  oi?  —  l  is  negative  for  all  real  values  of  x. 

21.  What  is  the  condition  that  aii? -\-hx-\- c  shall  have  the 
>;iuie  sign  for  all  real  values  of  -c? 


GENERAL   REVIEW 


441.    1.    Define    power;    root;    like   terms;    transposition; 
simultaneous  equations;  surd. 

2.  Distinguish  between  known  and  unknown  numbers. 

3.  Why  is  the  sign  of  multiplication  usually  omitted  be- 
tween letters,  and  never  omitted  between  figures  ? 

4.  How  is  multiplication  like  addition  ?   division  like  sub- 
traction ?     What  two  meanings  has  the  minus  sign  in  algebra  ? 

5.  When  X,  -T-,  or  both  occur  in  connection  with  +,  —  ,  or 
both  in  an  expression,  what  is  the  sequence  of  operations  ? 

6.  State  the  law  of  exponents  for  multiplication;  for  divi- 
sion. 

7.  When  is  a-"  —  y'^  divisible  by  both  x  -\-y  and  x  —  y? 

8.  When  is  a  trinomial  a  perfect  square  ?    'When  is  a  frac- 
tion in  its  lowest  terms  ?     What  are  similar  fractions  ? 

9.  What  operation  is  indicated  by  the  radical  sign?     In 
what  other  way  may  this  operation  be  indicated  ? 

10.  When  is  an  expression  both  integral  and  rational  ?  When 
are  expressions  said  to  be  prime  to  each  other  ? 

11.  By  what  principle  may  cancellation  be  used  in  reducing 
fractions  to  lowest  terms  ? 

12.  During  12  hours  of  a  certain  day,  the  following  tempera- 
tures were  recorded  at  Helena,  Montana :  —  9°,  —  8°,  —  8°, 
_  9  ,  _  9°.  _  9°,  _  8°,  +  12°,  +  25°,  -f  40°,  +  20°,  +  16°. 
Find  the  average  temperature  for  the  12  hours. 

360 


GENERAL   REVIEW  351 

13.  Detiiie  the  terms  conditional  equation ;  identical  equation. 

14.  Explain  the  meaning  of  a  negative  integral  exponent ;  of 
a  Iractional  exponent. 

15.  Define  evolution ;  radical;   entire  surd;  binomial  .surd; 
aiilar  surds. 

16.  Express  the  following  without  parentheses : 

{a'arr,  -  [-  (ayy,  (ay,  (ay. 

17.  What  is  meant  by  the  order  of  a  surd  ?  Illustrate  your 
answer  by  giving  surds  of  different  orders. 

18.  Tell  how  to  rationalize  a  binomial  quadratic  surd. 

19.  What  powers  of  V  —  1  are  real  ?  imaginary  ? 

20.  What  roots  should  be  associated  when  the  roots  of  a 
system  of  equations  are  given  thus:  x=  ±  2,  y=  T  3? 

21.  Illustrate  how  a  root  may  be  introduced  in  the  solution 
of  an  equation  ;  how  a  root  may  be  removed. 

22.  Why  is  it  specially  important  to  test  the  values  of  the 
unknown  number  found  in  the  solution  of  radical  equations? 

23.  Upon  what  axiom  is  clearing  equations  of  fractions 
based  ?  What  precautions  should  be  taken  to  prevent  intro- 
ducing roots?  If  roots  are  introduced,  how  may  they  be 
(N'tected? 

24.  Define  symmetrical  equation ;  quadratic  surd ;  coordinate 
axes;  imaginary  number;  axiom;  coefficient;  homogeneous 
polynomial ;  elimination. 

25.  Explain  how,  in  the  solution  of  problems,  negative  roots 
<'t  quadratic  equations,  while  mathematically  correct,  are  often 
inadmissible. 

26.  Define  negative  number,  subtraction,  and  multiplication, 
and  show,  from  your  definitioriy  how  the  following  rules  maybe 
•  i.'duced: 

(1)  "Change  the  sign  of  the  subtrahend  and  proceed  as  in 
addition ; " 

(2)  "  Give  the  product  the  positive  or  the  negative  sign  ac- 
rding  as  the  two  factors  have  like  or  unlike  signs." 


352  GENERAL   REVIEW 

27.  What  is  a  pure  quadratic  equation  ?  a  complete  quad- 
ratic equation?     Give  the  general  form  of  each. 

28.  State  two  methods  of  completing  the  square  in  the  solu- 
tion of  affected  quadratic  equations. 

29.  What  is  the  relation  between  the  factors  of  an  expression 
and  the  roots  of  the  equation  that  may  be  formed  by  putting 
the  expression  equal  to  0  ? 

30.  Outline  the  method  of  solving  quadratic  equations  by 
factoring. 

31.  Write  the  roots  of  the  quadratic  equation  ax'-\-  bx-i-c  =  0. 
Write  the  discriminant  of  the  equation.  What  relation  between 
the  coefficients  indicates  that  the  roots  are  imaginary? 
reciprocals  of  each  other  ? 

32.  What  is  the  advantage  of  letting  oc^  =  y  in  the  graphic 
solution  of  quadratic  equations  of  the  form  aa^  -{-  bx-j-c  =  0? 

33.  How  does  the  graph  of  a  quadratic  equation  show  the 
fact,  if  the  roots  are  real  and  equal  ?  real  and  unequal  ? 
imaginary  ? 

34  Prove  that  a  quadratic  equation  has  two  and  only  two 
roots 

35.  Tell  how  to  form  a  quadratic  equation  when  its  roots 
are  given.     Form  the  equation  whose  roots  are  |  and  ^. 

36.  What  is  the  meaning  of  "  function  of  x^'  ?  "  infinite 
number  "  ? 

37.  Tell  how  the  signs  of  the  roots  of  a  quadratic  equation 
may  be  determined  without  solving  the  equation. 

38.  Derive  the  value  of  the  sum  of  the  roots  of  the  equation 
aj2  _|_^^  _j_  ^  _() .  ^i^Q  value  of  the  product  of  the  roots. 

39.  In  clearing  a  fractional  equation  of  its  denominators, 
why  should  we  multiply  by  their  lowest  common  multiple? 

Illustrate   by   showing   what   happens   when   the   equation 
2x  10     ^     7 

X  —  1       0^—1       x  +  1 

is  multiplied  by  the  product  of  all  the  denominators. 


GENERAL   REVIEW  853 


EXERCISES 

442.    1.    Add  xy/y  +  yy/x-\- y/xy,  x^y^  —  y/¥y  —  Vx/,  Vxhf 
—  \xy'  —  Vxyy  and  y  Vx  —  x  V4^—  V9  xy. 

2.  What  number  must  be  subtracted  from  a  —  b  to  give 

b~<i  -\-c? 

3.  Simplify  a- )6-a-[a-6-(2a  +  6)-j-(2a-6)-a]-&S. 

4.  Multiply  a;V«-f-«V2/  +  2/Va;  +  ?/Vy  by  Va;— Vy. 

5.  Multiply  2  a;-'* -5  2/  -     by  2ar-*4-5y  *. 

6.  Expand  (a;*-2/'')(x*  +  3r)(a^  +  y^'*). 

7-    Divide  a^  —  t^  by  x  —  y  by  inspection.     Test. 

8.  Divide  ar*  —  3  ic*  —  20  by  a;  —  2,  by  detached  coefficients. 

9.  Show  by  the  factor  theorem  that  a^  —  6"  is  divisible  by 
x  +  b. 

10.  Divide  (a  +  6)  +  «  by  (a  +  b)^  +  xK 

11.  Factor  9 X*- 12a; +  4;  9x*4-9a;  +  2;  ar'-Sx-f  2;  a*  +  1. 

12.  Show  by  the  factor  theorem  that  x  —  a  is  a  factor  of 
X*  +  3  ax"-^  —  4  a\ 

13.  Separate  a"  —  1  into  six  rational  factors. 

14.  Factor4(ad  +  6c)2-(a«-6*^-c2  4-d*)«. 

15.  Find  the  H.C.F.  of  3x'-x-2  and  6x*-f  x-2. 

16.  Find  the  H.  C.  F.  of  4  a;*  - 11  x*  +  H  a?  - 12, 

2a^H-x»-4x»+7x-15,  and2a;*-f-»*-a;-12. 

17.  Find  the  L.  C.  M.  of  4  a*6x,  6  a6y,  and  2  axy. 

18.  Find  the  L.  C.  M.  of  x*  —  y*,  x  -f  y,  and  xy  —  y^. 

19.  Reduce  ^„~"     ~  ,^^ —  to  its  lowest  terms. 

.milnk'8  stand,  alo. — 23 


354  GENERAL  REVIEW 

T  X  Xi 

20.    Simplify -  — V— — -- 

^        x-\-l      1-x      X--1 


22.    Simplify 


x-\-y  x  —  y         j4  a^y 

2^     2x-\-2y      ?/ 

1  1 


21.    Simplify  ^^±^  +  -^-^^  +  4^. 
2x  —  2y      2x-\-2y      y^  —  x^ 


{a  -b){b-  c)      (c  -  h){c  -  a)      (c  -  a){a  -  b) 


23.  Simplify  i +  1 ^     ^     r-  —  a; j 

'     1+1  -+1|  ll-l  — 1 

[  X  }  I  oc  J 

24.  Simplify 

X — -      x4- 


X-\--  X 

•    X  X 

25.  Raise  a  —  b  to  the  seventh  power. 

26.  Expand  (2  a  +  3  5)^ ;    ( V^  +  Vyf ;    (- 1  -  V^'- 

27.  Find  the  sixth  root  of  4826809. 

28.  Reduce  Vf  to  its  simplest  form. 

29.  Reduce  -y/2Wa}  to  its  simplest  form. 

30.  Find  the  value  of  — n  to  3  decimal  places. 

V2 

31.  Multiply  2  + V8  by  1-V2;    2  +  V^^  by  1  -  a/^^ 

oo     c-       r-P     V3+3V2 

32.  Simplify  — — — 

^      V6  +  2 

33.  Show  that  (axf  =  1. 


a 


34.    Show  that  ax 

x" 


35.  Show  that  x^^  =  i/'x"^ ;    also  that  x^  =  {Vxf. 

36.  Find  the  value  of  125^ ;    of /"^V^  when  a;  =  .5. 


GENERAL   REVIEW 


355 


Solve  the  following  equations  for  x: 


37. 


1 


a-l-6 


a  —  b 


38. 


1        X 


a  +  b 

1 
X 


=0. 


40. 


mx-  —  nx 


or  -\ —  = • 

^2       2 


42.    Va;-9=V^-1. 


43.    a;2+V^Tl6  =  U. 


41.   ic«4-8  =  9x«. 

45.  (1  +  a:)*  +  (1  -  a:)*  =  242. 

46.  X  +  x^  -\-  (1  -^  x  +  x^' =  55. 

47     — _J-±4=^---  ^+^ 

l-|-a;+Vl  +  ar^ 

Solve  for  a;,  y,  and  z : 


+vrT^ 


48. 


49. 


50. 


51. 


52. 


^  +  ^  =  10, 

X     y 

X     y 


2x  +  32/  +  2  =  9, 
aj4-2y4-32  =  13, 
3« +  2/4-22  =  11. 

aj?H-.v-}-2  =  2(a  +  l), 
a;  +  y  +  02  =  a*  +  3. 


a^4-^7  =  24, 
/  +  in/  =  12. 


ra^  +  3a^  = 
lxj/  +  4y'  = 


18. 


53. 


54. 


55. 


56. 


57. 


58. 


59. 


'  ar'  +  x  =  26 
.a^  =  8. 

V^=12, 


y-Uy 


x-\-y  —Vx-\-y  =  20. 
(xy  -xy^=  -6, 
1  a;  —  a;^  =  9. 
{xy=x  +  y, 
lar^4../2^8. 
(x'f-^txy  =6, 
W  +  4y*  =  29. 
r2ar»  +  2y^  =  9a7/, 
lx4-y  =  3. 

ic^  +  yl  =4, 
a;^  +  ?/  =  16. 


356  GENERAL   REVIEW 

Problems 

443.  1.  How  far  down  a  river  whose  current  runs  3  miles  an 
hour  can  a  steamboat  go  and  return  in  8  hours,  if  its  rate  of 
sailing  in  still  water  is  12  miles  an  hour  ? 

2.  A  woman  on  being  asked  how  much  she  paid  for  her  eggs, 
replied,  "  Three  dozen  cost  as  many  cents  as  I  can  buy  eggs  for 
64  cents."     What  was  the  price  per  dozen  ? 

3.  A  man  had  not  room  in  his  stable  for  8  of  his  horses, 
so  he  built  an  additional  stable  |  the  size  of  the  other,  and 
then  had  room  for  8  horses  more  than  he  had.  How  many 
horses  had  he  ? 

4.  In  a  mass  of  copper,  lead,  and  tin,  the  copper  was  5 
pounds  less  than  half  the  whole  in  weight,  and  the  lead  and  tin 
each  5  pounds  more  than  i  of  the  remainder.  Find  the  weight 
of  each. 

5.  A  person  who  can  walk  n  miles  an  hour  has  a  hours  at  his 
disposal.  How  far  may  he  ride  in  a  coach  that  travels  m  miles 
an  hour  and  return  on  foot  within  the  allotted  time  ? 

6.  A  merchant  sold  half  a  car  load  more  than  half  his  grain  ; 
then  he  sold  half  a  car  load  more  than  half  the  remainder,  and 
then  found  that  if  he  should  sell  half  a  car  load  more  than  lialf 
of  what  he  still  had,  he  would  have  none  left.  How  many  car 
loads  of  grain  had  he  ? 

7.  A  man  received  $  2.50  per  day  for  every  day  he  worked, 
and  forfeited  $  1.50  for  every  day  he  was  idle.  If  he  worked 
3  times  as  many  days  as  he  was  idle  and  receivec^  $  24,  how 
many  days  did  he  work  ? 

8.  A  jeweler  has  two  silver  cups,  and  a  cover  worth  ^  1.50. 
The  first  cup  with  the  cover  on  it  is  w^orth  IJ  times  as  much  as 
the  second  cup,  and  the  second  cup  with  the  cover  on  it  is  worth 
11  as  much  as  the  first  cup.     Find  the  value  of  each  cup. 

9.  Find  two  numbers  such  that  their  sum,  their  product, 
and  the  difference  of  their  squares  are  all  equal. 


GENERAL   REVIEW  357 

10.  A  woman  has  13  squaxe  feet  to  add  to  the  area  of 

the  rug  she  is  weaving.  She  therefore  increases  the  length  | 
and  the  width  :J^,  which  makes  the  perimeter  4  feet  greater. 
Find  the  dimensions  of  the  finished  rug. 

11.  Twenty-eight  tons  of  goods  are  to  be  transported  in  carts 
and  wagons,  and  it  is  found  that  it  will  require  15  carts  and  12 
wagons,  or  else  24  carts  and  8  wagons.  How  much  can  each 
cart  and  each  wagon  carry  ? 

12.  There  is  a  number  whose  three  digits  are  the  same.  If 
7  times  the  sum  of  the  digits  is  subtracted  from  the  number, 
the  remainder  is  180.     What  is  the  number  ? 

13.  A  and  B  can  do  a  piece  of  work  in  m  days,  B  and  C  in 
u  days,  A  and  C  in  />  days.  In  what  time  can  all  together  do 
it?     How  long  will  it  take  each  alone  to  do  it? 

14.  Two  passengers  together  have  400  pounds  of  baggage 
and  are  charged,  for  the  excess  above  the  weight  allowed  free, 
40  cents  and  60  cents,  respectively.  If  the  baggage  had  be- 
longed to  one  of  them,  he  would  have  been  charged  $  1.50. 
How  much  baggage  is  one  passenger  allowed  without  charge  ? 

15.  It  took  a  number  of  men  as  many  days  to  pave  a  side- 
walk as  there  were  men.  Had  there  been  3  men  more,  the  work 
would  have  been  done  in  4  days.     How  many  men  were  there? 

16.  A  merchant  bought  two  lots  of  tea,  paying  for  both  $34. 
One  lot  was  20  pounds  heavier  than  the  other,  and  the  number 

of  cents  paid  per  pound  was  in  each  case  equal  to  the  number 
•  tf  pounds  bought.     How  many  pounds  of  each  did  he  buy  ? 

17.  A  and  B  hired  a  pasture  for  a  certain  sum  per  week.  A 
imt  in  4  horses,  and  B  as  many  as  cost  him  18  shillings  per 
week.  Later  B  put  in  2  additional  horses,  and  had  to  pay  20 
shillings  per  week.     Find  the  cost  of  the  pasture  per  week. 

18.  By  lowering  the  selling  price  of  apples  1  cent  a  dozen, 
a  man  finds  that  he  can  sell  60  more  than  he  used  to  sell  for 
60  cents.     At  what  price  per  dozen  did  he  sell  them  at  first  ? 


358 


GENERAL   REVIEW 


19.  A  railway  train,  after  traveling  2  hours  at  its  usual  rate, 
was  detained  1  hour  by  an  accident.  It  then  proceeded  at  f  of 
its  former  rate,  and  arrived  7|  hours  behind  time.  If  the  acci- 
dent had  occurred  50  miles  farther  on,  the  train  would  have 
arrived  6^  hours  behind  time.  What  was  the  whole  distance 
traveled  by  the  train  ? 

20.  A  and  B  left  Chicago  and  walked  in  the  same  direction 
at  uniform  rates,  B  starting  2  hours  after  A  and  overtaking 
him  at  the  30th  milestone.  Had  each  traveled  half  a  mile  more 
per  hour,  B  would  have  overtaken  A  at  the  42d  milestone.  At 
what  rate  did  each  travel  ? 

21.  The  load  on  a  wall  column  for  an  office  building  is 
360,000  pounds,  including  the  weight  of  the  column  itself,  and 


Interior 
Column 


Wall 
ColuiniT 


Girder  acting  as  a  lever 


i^isip? 


m 


is  balanced,  as  shown  in  the  figure,  by  a  part  of  the  load  on  an 
interior  column.  Neglecting  the  weight  of  the  girder,  find  the 
load  on  the  fulcrum. 

22.  A  projectile  fired  from  a  battleship  was  heard  by  the 
gunner  to  strike  a  mark  3360  feet  away  4i  seconds  after  it  was 
fired.  An  officer  on  another  vessel  5600  feet  from  the  first  and 
2240  feet  from  the  mark  heard  the  shot  strike  If  seconds 
before  the  report  reached  him.  Find  the  velocity  of  the  sound 
and  the  average  velocity  of  the  projectile. 

23.  The  velocity  acquired  or  lost  by  a  body  acted  upon  by 
gravity  is  given  by  the  formula  v  —  gt  (take  g  =  32.16).  If  a 
bullet  is  fired  vertically  upward  with  an  initial  velocity  of  2010 
feet  per  second,  in  how  many  seconds  will  it  return  to  the 
earth  (neglecting  the  friction  of  the  air)  ? 

Using  the  formula  s=l-gt-,  find  how  high  the  bullet  will  rise. 


GENERAL   REVIEW  859 

444.   The  following  are  from  recent  college  entrance  exami- 
nation papers : 

1.  Determine    graphically    the     roots     of    4  a;  +  5  y  =  24, 
:'>x  —  2y=  —5.     Give  the  construction  in  full. 

2.  Solve— 5— +  _1 z-T— I -=0. 

2x4-3      x-o      2a^-7x-lo 

3.  Solve  for  x  and  y,  (x  —  y)^  =  c^,  (y  —  a)(x  —  6)  =  0. 

4.  Find  X  from  the  equations, 

fxr-^xy-{-z  =  2y 
x-\-2y-\-z  =  S, 
x  —  y-\-z  =  0. 
Sr<;GKSTiox.  — From  the  second  equation  subtract  the  third. 

5.  Solve  (2(^  +  2/)^-(^  +  ^)(^-22/)  =  70, 

[2(x  +  y)-3{x^2y)=2. 

6.  Solve  for  x,  y,  and  z, 

(x-\-y  =  xy, 

\2x  +  2z  =  xz, 

[sz-^Sy  =  yz. 

SnooESTiON. — Find  y  in  terms  of  x,  and  z  in  terms  of  z;  substitute 
these  results  in  the  remaining  equation. 


7.    Solve  a;-y- V^-y  =  2,  0^-2/3  =  2044. 


8.    Solve  for  x  and  y  , 


(a  H-  c)x  —  (a  —  c)y  =  2  ahy 
(a  +  b)x  —  {a  —  b)y  =  2  ac. 
9.    Factor  4 aJ* 4-1;  27a^4-3a;-2;  ^  x*  +  y*  - T)  ary\ 

10.  Resolve  into  prime  factors : 
3(a_l)s_(l_}_a);  a*-a'b^-b'-l;  3  x'^ -\- 7  x'^  -  6. 

11.  Solve  the  equation  4  ar  H-  mx  4-5  =  0.     For  what  values 
of  m  are  the  roots  imaginary  ? 

12.  How  much  water  must  be  added  to  80  pounds  of  a  6  % 
salt  solution  to  obtain  a  4  %  solution  ? 


360  GENERAL   REVIEW- 

IS.    Construct  the  graph  of  the  function  x^  —  2  x-\-l. 

14.  Under  what  conditions  will  the  roots  of  ax^  -\-bx-\-c  =  0 
be  positive  ?  negative  ?  one  positive  and  the  other  negative  ? 

15.  Find  to  four  terms  the  square  root  of  ic^  —  3  a?  +  1. 

16.  Find  the  square  root  of 

9  a'c""'  _  3  ao"'+"  ,  ,6  ,n     2«ac«  ,  2^6V  ,  2^« 


17.  8olYex--i-7x-S  =  ^2x'-\-Ux-{-2. 

18.  Simplify  &<^>  -  c)  +  acjc  -  a)  +  a^a  -  ft) 

a^  -\-bG  —  ac  —  ab 


19.    Solve  < 


3       12      1^2        3  J         .3' 
X  —  ?/     1 


20.    Solve 


.x-{-y     5 

Vic  —  Vy  =  2, 
(Va?—  V^/)  Va;?/ =  30. 

21.  Show  by  the  factor  theorem  that  a"  +  6"  is  exactly 
divisible  by  a  +  6  for  all  positive  odd  integral  values  of  n. 

22.  The  area  of  the  floor  of  a  room  is  120  square  feet ;  of 
one  end  wall,  80  square  feet ;  and  of  one  side  wall,  96  square 
feet.     Find  the  dimensions  of  the  room. 

23.  Show  that  V3  is  greater  than  V6. 

a.-  +  /  =  4, 


24.  Draw  the  graphs  of  the  two  equations  , 

and  tell  the  algebraic  meaning  of  the  fact  that  the  two  graphs 
do  not  intersect. 

25.  A  rectangular  piece  of  tin  is  4  inches  longer  than  it 
is  wide.  An  open  box  containing  840  cubic  inches  is  made  from 
it  by  cutting  a  6-inch  square  from  each  corner  and  turning  up 
the  ends  and  sides.     What  are  the  dimensions  of  the  box  ? 


INEQUALITIES 


445.  Any  problem  thus  far  presented  has  been  such  that 
its  conditions  could  be  stated  by  means  of  one  or  more  equa- 
tions. In  some  problems  and  exercises,  however,  the  condi- 
tions are  such  as  to  lead  to  a  statement  that  one  number  is 
(jreater  or  less  than  another.  It  is  the  purpose  of  this  chapter 
to  discuss  such  statements,  for  they  often  yield  all  the  data 
necessary  to  the  required  solution. 

446.  One  number  is  said  to  be  greater  than  another  when  the 
remainder  obtained  by  subtracting  the  second  from  the  first  is 
positive^  and  to  be  less  than  another  when  the  remainder 
obtained  by  subtracting  the  second  from  the  first  is  negative. 

If  a  —  b  is  a  positive  number,  a  is  greater  than  b ;  but  if  a  —  6  is  a 
negative  number,  a  is  less  than  h. 

Any  negative  number  is  regarded  as  less  than  0 ;  and,  of  two 
negative  numbers,  that  more  remote  from  0  is  the  less. 

Thus,  —  1  is  less  than  0  ;  —  2  is  less  than  —  1 ;  —  3  is  less  than  —  2 ;  etc. 

An  algebraic  expression  indicating  that  one  number  is  greater 
or  less  than  another  is  called  an  inequality. 

447.  The  sign  of  inequality  is  >  or  <. 

It  is  placed  between  two  unequal  numbers  with  the  opening 
toward  the  greater. 

'a  t«  greater  thanh''  is  written  a>h;  'a  is  /esx  than  ft'  is  written 
<i<b. 

The  expressions  on  the  left  and  right,  respectively,  of  the 
sign  of  inequality  are  called  the  first  and  the  second  members  of 
the  inequality. 

861 


362 


INEQUALITIES 


448.  The  signs  >  and  <  are  read,  respectively,  'is  not 
greater  than  '  and  '  is  not  less  than.' 

449.  When  the  first  members  of  two  inequalities  are  each 
greater  or  each  less  than  the  corresponding  second  members, 
the  inequalities  are  said  to  subsist  in  the  same  sense. 

When  the  first  member  is  greater  in  one  inequality  and  less 
in  another,  the  inequalities  are  said  to  subsist  in  a  contrary 
sense. 

X  >  a  and  y^b  subsist  in  the  same  sense,  also  x  <  3  and  y  <  4  ;  but 
x'>  b  and  t/ <  a  subsist  in  a  contrary  sense. 

450.  The  following  illustrate  operations  with  inequalities : 


3. 


5. 


Given               8  >  5 

2. 

Given 

8>5 

Add,                 2      2 

Subtract, 

§" 

2      2 

84-2>5  +  2 

-  2  >  5  -  2 

That  is,          10  >  7. 

That  is. 

6>3. 

Given                8  >  5 

4. 

Given 

16>10 

Multiply,          2      2 

Divide, 

2      2 

8.2>5.2 

16 

-2>10--2 

That  is,          16  >  10. 

That  is. 

8>5. 

Given                8  >  5 

6. 

Given 

16>10 

Multiply,      —  2       —  2 

Divide, 

-2       -2 

8._2<5.-2 

16 

--■ 

_2<10--2 

That  is,      —  16  <  —  10. 

That  is. 

-8<-5. 

451.  Principle  1.  —  If  the  same  number  or  equal  numbers 
he  added  to  or  subtracted  from  both  members  of  an  inequality, 
the  resulting  inequality  ivill  subsist  in  the  same  sense. 

For,  let  n>h,  and  let  c  be  any  positive  or  negative  number. 

Then,  §146,  a  —  b  =  p,  a,  positive  number. 

Adding  c  —  c  =  0,  Ax.  1,     «  +  c  —  (b  +  c)  =  p. 

Therefore,  a  +  c*  >  6  +  c. 

Note.  —  Letters  used  in  this  chapter  stand  for  real  numbers. 


INEQUALITIES  363 

452.  Principle  2.  —  If  both  members  of  an  inequality  are 
multiplied  or  divided  by  the  same  number ^  the  resulting  inequality 
will  subsist  in  the  same  sense  if  the  multiplier  or  divisor  is  j)osi- 
ticej  but  in  the  contrai^  sense  if  the  multiplier  or  divisor  is 
nefjative. 

For,  let  a>b. 

Tlieii,  §  446,  a  —  b=p,  a  positive  number. 

Multiplying  by  m,  ma  —  mh  =  mp. 

If  m  is  positive,  mp  is  positive, 

and  therefore,  §  446,  ma  >  mb. 

If  m  is  negative,  mp  is  negative, 

and  therefore,  §  446,  ma  <  mb. 

Putting  —  for  to,  the  principle  holds  also  for  division, 
m 

453.  Principle  3.  —  A  term  rtiay  be  transposed  from  one 
member  of  an  inequality  to  the  other,  provided  its  sign  is  changed. 

For,  let  a  —  b>c  —  d. 

Adding  b  to  each  side,  Prin.  1,  a>b  +  c  —  d. 

Adding  -  c  to  each  side,  Prin.  1,    a-  c>b-d. 

454.  Principle  4.  —  If  the  signs  of  all  the  terms  of  an  in- 
etjmdity  are  changed,  the  resulting  inequality  will  subsist  in  the 
contrary  sense. 

For,  let  a-b>c  -d. 

Multiplying  each  side  by  —  1,  Prin.  2, 

-a  +  h<  -c  +  d. 

455.  Principle  5.  —  If  the  corresponding  members  of  any 
number  of  inequalities  subsisting  in  the  same  sense  are  added, 
the  resulting  inequality  will  subsist  in  th£  same  sense. 

For,  let  a>bj  c>d,  e >/,  etc. 

Then,  §  446,  a  —b,  c  —  d,  e  —f  etc.,  are  positive. 

Hence,  their  sum,  a  +  c  +  <?  + (6  +  rf  +/+•••)>  is  positive; 

that  is,  a  +  c  +  e-i — >^ +  </+/+•••• 


864  INEQUALITIES 

Note.  —  The  student  should  bear  in  mind  that  the  difference  of  two 
inequalities  subsisting  in  the  same  sense,  or  the  sum  of  two  inequalities 
subsisting  in  a  contrary  sense,  may  have  its  first  member  greater  than, 
equal  to,  or  less  than  its  second. 

Thus,  take  the  inequality  12  >  6. 

Subtracting  7  >  3,  or  adding  —  7  <  —  3,  the  result  is  5  >  3. 
Subtracting  8  >  2,  or  adding  —  8  <  —  2,  the  result  is  4  =  4. 
Subtracting  8  >  1,  or  adding  —  8  <  —  1,  the  result  is  4  <  6. 

456.  Principle  6.  —  If  each  member  of  an  inequality  is  sub- 
tracted from  the  corresponding  member  of  an  equation^  the  result- 
ing inequality  will  subsist  in  the  contrary  sense. 

For,  let  a  >  6  and  let  c  be  any  number. 
Then,  §  446,  a  —  b  is  a  positive  number. 

Since  a  number  is  diminished  by  subtracting  a  positive  number 
f^*o"^  it,  c-(a-b)<c. 

Transposing,  c  —  a  <Cc  —  b. 

That  is,  if  each  member  of  the  inequality  a>ft  is  subtracted  from 
the  coiresponding  member  of  the  equation  c  =  c,  the  result  is  an 
inequality  subsisting  in  a  contrary  sense. 

457.  Principle  7.  — If  a  >  b  and  b  >  c,  then  a  >  c. 
For,  §  446,  a  —  h  is  positive  and  b  —  c  is  positive. 
Therefore,  (a  —  b)  +  (b  —  c)  is  positive ; 

that  is,  simplifying,  a  —  c  is  positive. 

Hence,  §  446,  a  >  c. 

Note.  —  In  a  similar  manner,  it  may  be  shown  that  if  a<  6  and  & <  c, 
then  a  <  c. 

458.  Principle  8.  —  If  the  corresponding  members  of  two 
inequalities  subsisting  in  the  same  sense  are  multiplied  together, 
the  result  icill  be  an  inequality  subsisting  in  the  same  sense,  pro- 
vided all  the  members  are  positive. 

For,  let  a>6  and  c'>d,  a,  b.  c,  and  d  being  positive. 
Multiplying  the  first  inequality  by  c  and  the  second  by  b,  Prin.  2, 

ac  >  be  and  be  >  bd. 
Hence,  Prin.  7,  ac>bd. 


INEQUALITIES  865 

Notes.  —  1.  When  some  of  the  members  are  negative,  the  result  may 
be  an  inequality  subsisting  in  the  same  or  in  a  contrary  sense,  or  it  may 
be  an  e(iuation. 

Thus,  take  the  inequality  12  >6. 

Multiplying  by  -  2  >  -  6,  -  2  >  -  3,  and  -  2  >  -  4,  the  results  are, 
respectively,  -  24  >  -  30,  -  24  <  -  18,  and  -  24  =  -  24. 

2.  The  quotient  of  tvfo  inequalities,  member  by  member,  may  have  its 
first  member  greater  than,  equal  to,  or  less  than,  its  second. 

Thus,  take  the  inequality  12  >  6. 

Dividing  by  3  >  2,  4  >  2,  and  —  2  >  —  3,  tlie  results  are,  respectively, 
4  > 3,  3  =  3,  and  -6<  -2. 

EXERCISES 

459.    1.   Find  the  values  of  x  in  the  inequality  3  x  —  10  >  11. 

Solution 

3a;-10>ll. 

I»rin.  lor3,  3x>2l. 

Prin.  2,  z>T. 

Therefore,  for  all  values  of  x  greater  than  7,  the  inequality  is  true ;  that 
Is,  the  inferior  limit  of  a;  is  7. 

2.  Find  the  values  of  x  in  the  simultaneous  inequalities 
3  .f  -f-  5  <  38  and  4  a;  <  7  a;  —  18. 

SOUTION 

3x  +  5<:i8.  (1) 

4x<7x-18.  (2) 

Transposing  in  (1),  Prin.  3,  3  x  <  33. 

.-.  Prin.  2,  x<ll. 

Transposing  in  (2),  Prin.  .S,      -  3  x  <  —  18. 

.-.Prin.  2,  x>6. 

The  result  shows  that  the  given  inequalities  are  satisfied  simultaneously 
by  any  value  of  x  between  6  and  11 ;  that  i.s.  thr  ,- /.  ;  ,,•  limit  of  x  is  6, 
and  the  superior  limit  11. 


366  INEQUALITIES 

Find  the  limits  of  x  in  each  of  the  following : 

3.  6a;-5>13.  (4x-ll>^x, 

4.  5a;-l<6a;  +  4.  *     l20-2a;>10. 

5.  3x-ia7<30.  r3_4a;<7, 

6.  4:X-^l<6x-ll.  [ox 
2  a;  ,  5  x . 


9.    a;  +  ^  + 

3         6 


+  10<20. 
>  25  and  <  30. 


10.    Find  the  limits  of  a?  and  y  in  3a;— 2/>— 14andic-f2  ?/=0. 
Solution 

(Sx-y>-14,  (1) 

\x  +  2y  =  0.  (2) 

6x-2y>-2S.  (8) 
7x>-28. 

x>-i.  (4) 

3  X  +  6  1/  =  0.  (5) 
1y>-U. 


Multiplying  (1)  by  2, 
Adding  (2)  and  (3), 
Dividing  by  7, 
Multiplying  (2)  by  3, 
Subtracting  (5)  from  (1), 


Dividing  by  —  7,  y<2. 

That  is,  X  is  greater  than  —  4,  and  y  is  less  than  2. 

Find  the  limits  of  x  and  y  in  the  following,  and,  if  possible, 
one  positive  integral  value  for  each  unknown  number : 


11. 


12. 


13. 


2x-Sy<2, 
2x-\-5y  =  25. 

3x  +  2y=^4:2, 

3a;--^>16. 

7 

f  a;  -f-  ?/  =  10, 
4:X<3y. 


14. 


15. 


16. 


y  =  Sx  +  4t, 
^2o<2y+Sx. 

i^y-x>9, 

[  20      15 

|a;>2/4-4, 
[x-2y  =  8.     . 


INKQUALITIES  367 

17.  Find  the  limits  ot  x  in  x^  +  3  x  >  28. 

Solution 

a:*  +  3  X  >  28. 

Transposing,  Prin.  3,         x-  +  3  x  —  28  >  0. 
Factoring,  (x  —  4) (x  +  7)  >  0. 

rhat  is,  (x  —  4)  (x  +  7)  is  positive. 

Since  (x— 4)  (x  +  7)  is  positive,  either  both  factors  are  positive  or 
both  are  negative.  Both  factors  are  positive  when  x  >  4  ;  both  factore 
are  negative  when  x<  —  7. 

Hence,  x  can  have  all  values  except  4  and  —  7  and  intermediate  values. 

Find  the  limits  of  x  in  each  of  the  following : 

18.  a^'-h3a;>10.  22.   x^  >  9  x  -  18. 

19.  x2-f8a;>20.  23.    a:^  +  40  a;  >  3  (4  .-c  -  25). 

20.  3T  -^6x>  24.  24.    x^  -\- bx  >  ax -\-  ab. 

21.  (x-2)(S-x)>0.  25.    (a;  -  3)(5  -  a;)  >  0. 

26.  If  a  and  b  are  positive  and  unequal,  prove  that 

a^-f //>2a6. 

Proof 

Whether  a  —  ft  is  positive  or  negative,  (a  —  h)'^  is  positive ;  and  since 
(I  and  h  are  unequal,  (a  —  b)'^  >  0; 

tliatis,  a^-2ab  +  b^>0. 

Transposing,  Priii.  :i       a^  +  h'^  >  2  ab. 

XoTE.  —  If  a  =  h,  it  is  evident  that  a*  +  6"^  =  2  ab. 

27.  When  a  and  b  are  positive,  which  is  the  greater, 

a-\-2b      aH-36  ' 


368  INEQUALITIES 

If  a,  b,  and  c  are  positive  and  unequal : 

28.  Prove  that  cr  -f  b'-  +  c"-'  >  ab  +  ac  -|-  be. 

29.  Prove  that  cr  +  ?/  >  a-/>  -f  ab'-. 

30.  Which  is  the  greater/^^  "^  ^  or  ^^'^  +  ^^? 

a  +  /j       f/,-  -f-  b'- 

31.  Prove  that  —  +  --->!,  except  when  2  a  =  3  6. 

lib      4:  a 

32.  Prove  that  (a  —2b)(4:b  —  a)<  b%  except  when  a  =  3  6. 

33.  Prove  that  a'  +  b^  +c^>3  abc. 

34.  Prove  that  tlie  sum  of  any  positive  real  number  (except 
1)  and  its  reciprocal  is  always  greater  than  2. 

35.  Prove  that  a  positive  proper  fraction  is  increased  by 
adding  the  same  positive  number  to  each  of  its  terms. 

36.  Find  the  smallest  whole  number  such  that  ^  of  it 
decreased  by  1  is  greater  than  i  of  it  increased  by  3. 

37.  If  5  times  the  number  of  pupils  in  a  certain  department, 
plus  25,  is  less  than  6  times  the  number,  minus  74 ;  and  if  twice 
the  number,  plus  50,  is   greater   than   3    times    the   number,  j 
minus  51,  how  many  pupils  are  there  in  the  department  ? 

38.  At  least  how  many  dollars  must  A  and  B  each  have 
that  5  times  A's  money,  plus  B's  money,  shall  be  more  than 
$  51,  and  3  times  A's  money,  minus  B's  money,  shall  be  $  21  ? 

39.  Four  times  the  number  of  passenger  trains  entering  a 
certain  city  daily,  minus  136,  is  less  than  three  times  the  num- 
ber, plus  24 ;  and  4  times  the  number,  plus  63,  is  less  than  5 
times  the  number,  minus  95.  How  many  passenger  trains 
enter  the  city  each  day  ? 

40.  Three  times  the  number  of  soldiers  in  a  full  regiment, 
less  593,  is  less  than  2  times  the  number,  plus  608 ;  and  8  times 
the  number,  minus  577,  is  less  than  9  times  the  number,  minus 
1776.     How  many  soldiers  are  there  in  a  full  regiment  ? 


RATIO   AND   PROPORTION 


RATIO 

460.  The  relation  of  two  numbers  tliat   is  expressed  by  the 
luotinit  of  the  first  divided  by  the  second  is  called  their  ratio. 

461.  i'he  sign  of  ratio  is  a  colon  (:). 

A  ratio  is  expressed  also  in  the  form  of  a  fraction. 

a 

b' 

The  colon  is  sometimes  regarded  as  derived  from  the  sign  of  division 
by  omitting  the  line. 

462.  To  compare  two  quantities  they  must  be  expressed  in 
mis  of  a  common  unit. 

Thus,  to  indicate  the  ratio  of  20  f  to  $  1,  both  quantities  must  be  ex- 
pressed either  in  cents  or  in  dollars,  as  20*  :  100^  or  -S^  :  ^1. 
There  can  be  no  ratio  between  2  lb.  aim  ;;  it. 

The  ratio  of  two  quantities  is  the  i-'iHn  oi  their  numerical 
iiteasures. 

Thus,  the  ratio  of  4  rods  to  6  rods  is  the  ratio  of  4  to  5. 

463.  The  first  term  of  a  ratio  is  called  the  antecedent,  and 
the  second,  the  consequent.     Both  terms  form  a  couplet. 

The  antecedent  corresponds  to  a  dividend  or  numerator;  the 
•  onsequent,  to  a  divisor  or  denominator. 

In  the   ratio  a  :  />,  or  - ,  a  is  the  antecedent,  h  the  consequent,  and 
h 
tht'  terms  a  and  h  form  a  couplet. 

milne's  stand,  alg.  —  iM  '^r^^ 


370  RATIO   AND   PKOPORTION 

464.  A  ratio  is  said  to  be  a  ratio  of  greater  inequality,  a  ratio 
of  equality,  or  a  ratio  of  less  inequality,  according  as  the  ante- 
cedent is  greater  than,  equal  to,  or  less  than  the  consequent.' 

Thus,  when  a  and  b  are  positive  numbers,  a  :  b  is  a,  ratio  of  greater 
inequality,  if  a  >  6  ;  a  ratio  of  eqiiaUty,  it  a  =  b  ;  and  a  ratio  of  less 
inequality,  if  a<.b. 

465.  The  ratio  of  the  reciprocals  of  two  numbers  is  called 
the  reciprocal,  or  inverse,  ratio  of  the  numbers. 

It  may  be  expressed  by  interchanging  the  terms  of  the 
couplet. 

The  inverse  ratio  of  a  to  6  is  -  :  - .     Since  -  -^     =  -,  the  inverse  ratio 
,  a      b  aba 

of  a  to  6  may  be  written  -,  or  b  :  a. 
a 

466.  The  ratio  of  the  squares  of  two  numbers  is  called  their 
duplicate  ratio;  the  ratio  of  their  cubes,  their  triplicate  ratio. 

The  duplicate  ratio  of  a  to  6  is  a^  :h^  ;  the  triplicate  ratio,  a^  :  b^. 

467.  If  the  ratio  of  two  numbers  can  be  expressed  by  the 
ratio  of  two  integers,  the  numbers  are  called  commensurable 
numbers,  and  their  ratio  a  commensurable  ratio. 

468.  If  the  ratio  of  two  numbers  cannot  be  expressed  by 
the  ratio  of  two  integers,  the  numbers  are  called  incommen- 
surable numbers,  and  their  ratio  an  incommensurable  ratio. 

The  ratio  v'2  :  3  =  -^—  =    '      — '- —  cannot  be  expressed  by  any  two 

integers,  because  there  is  no  number  that,  used  as  a  common  measure, 
will  be  contained  in  both  V2  and  3  an  integral  number  of  times.  Hence, 
\/2  and  3  are  incommensurable,  and  \/2  :  3  is  an  incommensurable  ratio. 
It  is  evident  that  by  continuing  the  process  of  extracting  the  square 
root  of  2,  the  ratio  V2  :  3  may  be  expressed  by  two  integers  to  any 
desired  degree  of  approximation,  but  never  with  absolute  accuracy. 


Properties  of  Ratios 

469.  It  is  evident  from  the  definition  of  a  ratio  that  ratios 
have  the  same  properties  as  fractions ;  that  is,  they  may  be 
redvA^ed  to  higher  or  lower  terms,  added,  subtracted,  etc.     Hence, 


RATIO  AND   PROPORTION  371 

Principles.  —  1.  Mnltij^lying  or  dirl<ii>><i  i»,tJ>  lerms  of  a  ratio 
hii  the  same  number  does  not  chan'i>'  th<'  rtih/,'  of  the  ratio. 

2.  Multiplying  the  antecedent  or  dlcidiny  the  consequent  of  a 
ratio  by  any  number  multiplies  the  ratio  by  that  number. 

3.  Dividing  the  antecedent  or  multiplying  the  consequent  by 
any  number  divides  the  rcfio  by  that  number. 

470.  If  the  same  positive  number  is  added  to  both  terms  of 
;i  fiiictiou,  the  vahie  of  the  fraction  will  be  nearer  1  than  before, 
whether  the  fraction  is  improper  or  proper.  The  correspond- 
ing principle  for  ratios  follows : 

l*iMNriPLE  4.  —  A  ratio  of  greater  inequality  is  decreased  and 
(I  ratio  of  less  inequality  is  increased  by  adding  the  same  positive 
n  umber  to  ea/ih  of  its  terms. 

For,  given  the  positive  numbers  a,  6,  and  c,  and  the  ratio  -  • 

1 .  When  «  >  6,    it  is  to  be  proved  that   <  -  • 

b  +  c      b 

a  +  c      a  _c(h  —  n) 
b  +  c      b~  b{b  +  c) 

Since  6  —  a  is  negative,  because  a  >  6, 

^^^'T  ^)  is  negative ; 
b(h  +  c) 

therefore,  a  +  c  _  a  .^  negative. 

b-\-c      b 

Hence,  §446,  ^^<^- 

b  +  c     b 

2.  When  a  <  6,  it  is  to  be  proved  that  ^ >  ^  • 

b  +  c       b 

Proceeding  by  the  method  used  in  1,  since  a  <6  it  may  be  shown 

that  *LZ-^  —  -  is  positive. 

6  +  c      b 

Hence,  §140,  ?Lil£>2. 

b  -k-  c      b 


372  RATIO   AND  PROPORTION 

EXERCISES 

471.    1.    What  is  the  ratio  of  8  m  to  4m?  of  4  m  to  8  m  ? 

2.  Express  the  ratio  6  :  9  in  its  lowest  terms;    the  ratio 
12  a; :  16  2/ ;  am:  bin ;  20  ah  :  10  he ;  {m  +n )  :  {m~  —  iv). 

3.  Which  is  the  greater  ratio,  2:3  or  3:4?  4:9  or  2:5? 

4.  What  is  the  ratio  of  i  to  J  ?  i  to  i  ?  |  to  f  ? 
Suggestion.  —  When  fractions  have  a  common  denominator,  they  have 

the  ratio  of  their  numerators. 

5.  What  is  the  inverse  ratio  of  3  :  10  ?  of  12  :  7  ? 

6.  Write  the  duplicate  ratio  of  2  :  3 ;  of  4  :  5 ;  the  triplicate 
ratio  of  1 :  2  ;  of  3  :  4. 

Reduce  to  lowest  terras  the  ratios  expressed  by : 


7.    10:2. 

10.    3:27. 

13. 

«• 

16.   75-100. 

8.    12:6. 

11.    4:40. 

14. 

if- 

17.   60-120. 

9.    16:4. 

12.   9:72. 

15. 

«. 

18.    80-240. 

19.  What  is  the  ratio  of  15  days  to  30  days  ?  of  21  days  to 
1  week?  of  1  rod  to  1  mile  ? 

Find  the  value  of  each  of  the  following  ratios : 

20.  ^x:ty?.  23.    2| :  7f  26.    a-h^x' '.  a'h'X^. 

21.  \ah:\aG.        24.    .7m:.8  7i.      27.    {x- -  if)  :  {x  -  yf. 

22.  ^x^y^:\xy.     25.    A:i9:10x\    28.    (a'^- 1)  :  (cr  +  a+l). 

29.  Two  numbers  are  in  the  ratio  of  4  :  5.     If  9  is  subtracted 
from  each,  find  the  ratio  of  the  remainders. 

30.  Change  each  to  a  ratio  whose  antecedent  shall  be  1 : 

5:20;    3i«:12a;;    f:|;     .4:1.2. 

31.  Reduce  the  ratios  a  :  h  and  x:y  to  ratios  having  the 
same  consequent. 

32.  When  the  antecedent  is  6  a;  and  the  ratio  is  \,  what  is 
the  consequent  ? 


RATIO  AND  PROPORTION  373 

33.  CTiven  the  ratio  f  and  a  positive  number  x.     Prove  that 

"  "^     >  -  by  subtracting  one  ratio  from  the  other. 
3  -f"^        3 

Suggestion.  —  Proceed  as  in  §  470. 

34.  The  capital  stock  of  a  street  railway  company  was 
$7,500,000,  the  gross  earnings  for  a  year  $1,500,000,  and  the 
net  earnings  $600,000.  Find  the  ratio  of  gross  earnings  to 
capital  stock;  of  net  earnings  to  gross  earnings;  of  net  earn- 
ings to  capital  stock. 

PROPORTION 

472.  An  equality  of  ratios  is  called  a  proportion. 

8  :  10  =  6  :  20  and  a  :a;  =  6  :  y  are  proportions. 

The  double  colon  (: :)  is  often  used  instead  of  the  sign  of 
equality. 

The  double  colon  has  been  supposed  to  repre.sent  the  extremities  of 
the  lines  that  fonn  the  sign  of  equality. 

The  proportion  a :  &  =  c :  rf,  or  a:b:  :c:(l,  is  read,  *  the  ratio 
of  a  to  6  is  equal  to  the  ratio  of  c  to  d,'  or  *a  is  to  6  as 
'•  is  to  d.' 

473.  In  a  proportion,  the  first  and  fourth  terms  are  called 
tlie  extremes,  and  the  second  and  third  terms,  the  means. 

In  a  :  h  =  c  :  d^  a  and  d  are  the  extremes,  h  and  c  are  the  means. 

474.  Since  a  proportion  is  an  equality  of  ratios  each  of 
which  may  be  expressed  as  a  fraction,  a  proportion  may  be 
expressed  as  an  equation  each  member  of  which  is  a  fraction. 

Hence,  it  follows  that : 

General  Principle.  —  Tfie  changes  that  may  he  made  in  a 
ItropoHion  toithout  destroyiiifj  the  equality  of  its  ratios  correspond 
to  the  changes  that  may  be  made  in  the  members  of  an  equation 
ii'ithout  destroying  their  equality  and  m  the  terms  of  a  fraction 
nithout  altering  the  value  of  the  fraction. 


374  RATIO   AND   PROPOKTION 

Properties  of  Proportions 

475.  Principle  1.  —  In  any  jrroportion,  the  product  of  the  ex- 
tremes is  equal  to  the  product  of  the  means. 

For,  given  a:b  =  c:d, 

a      c 

Clearing  of  fractions,  ad  —  be. 

Test  the  following  by  principle  1  to  find  whether  they  are 
true  proportions : 

1.    6:16  =  3:8.  2.    11  =  if  3.    7:8  =  10:12. 

476.  In   the   proportion   a:m=m:b,  m   is    called   a  mean 
proportional  between  a  and  b. 

By  Prin.  1,  711^  =  ab; 

.'.  m=  Vab. 

Hence,  a  mean  proportioyial  between  two  numbers  is  equal  to 
the  square  root  of  their  product. 

1.  Show  that  the  mean  proportional   between  3  and  12  is 
either  6  or  —  6.     Write  both  proportions. 

2.  Find  two  mean  proportionals  between  4  and  25. 

477.  Principle  2.  —  Either  extreme  of  a  proportion  is  equal 
to  the  product  of  the  means  divided  by  the  other  extreme. 

Either  mean  is  equal  to  the  product  of  the  extremes  divided  by 

the  other  mean. 

For,  given  a:b  =  c:d. 

By  Prin.  1,  ad  =  be. 

Solving  for  a,  c?,  &,  and  c,  in  succession,  Ax.  4, 

be    J      be    -,      ad  ad 

a  =  -,d=^,b='-,c=—- 
a  a  c  0 

1.  Solve  the  proportion  3  :  4  =  x :  20,  for  x. 

2.  Solve  the  proportion  a; :  «  =  2  7n :  n,  for  x. 


RATIO    AND   PROPORTION  375 

3.  If  a  :  6  =  6  :  c,  the  term  c  is  called  a  third  proportional  to  a 
and  b.     Find  a  third  proportional  to  6  and  2. 

4.  In  the  proportion  a:b  =  c:d,  the  term  d  is  called  a  fourth 
proportional  to  a,  6,  and  c.  Find  a  fourth  proportional  to  |,  J, 
and  i- 

478.  Principle  3.  —  If  the  prodtict  of  two  numbers  is  equal 
to  the  product  of  two  other  numbers^  one  pair  of  them  may  be 
made  the  extremes  and  the  other  pair  the  means  of  a  proportion. 

For,  given  ad  =  he. 

Dividing  by  ftrf,  Ax.  4,  7  =  ^5 

h      a 

that  is,  a:h  =  c:d. 

By  dividing  both  members  of  the  given  equation,  or  of  be  =  ad,  by 
the  proper  numbers,  various  proportions  may  be  obtained ;  but  in  all 
f  them  a  and  d  will  be  the  extremes  and  b  and  c  the  means,  or  vice 
•  ■>  rsQy  as  illustrated  in  the  proofs  of  principles  4  and  5. 

1.  If  a  men  can  do  a  piece  of  work  in  x  days,  and  if  b  men 
(!an  do  the  same  work  in  y  days,  the  number  of  days'  work  for 
one  man  may  be  expressed  by  either  ax  or  by.  Form  a  pro- 
portion between  a,  b,  x,  and  y. 

2.  The  formula  pd  =  WD  (See  p.  173) 

expresses  the  physical  law  that,  when  a  lever  just  balances, 
tlie  product  of  the  numerical  measures  of  the  power  and  its 
distance  from  the  fulcrum  is  equal  to  the  product  of  the 
numerical  measures  of  the  weight  and  its  distance  from  the 
fulcrum.     Express  this  law  by  means  of  a  proportion. 

479.  Principle  4.  —  If  four  numbers  are  in  proportion^  the 
ratio  of  the  antecedents  is  equal  to  the  ratio  of  the  consequents; 
t  hat  is,  the  numbers  are  iij.  proportion  by  alternation. 

For,  given  a:b  —  c:d. 

Then,  Prin.  1,  ad  =  bc. 

Dividing  by  cd,  Ax.  4,  -  =  -  ; 

e      d 

tliat  i-.  a:c  =  b:d. 


376  RATIO   AND   PROPORTION 


480.  Principle  5.  —  If  four  numbers  are  in  proportion,  the 
ratio  of  the  second  to  the  first  is  equal  to  the  ratio  of  the  fourth 
to  the  third;  that  is,  the  numbers  are  in  proportion  by  inversion. 

For,  given  a  :b  =  c  :d. 

Then,  Prin.  1,  ad  =  be. 

.'.  be  =  ad. 

Dividing  by  ac,  Ax.  4,  -  =  -  ; 

a      c 

that  is,  b:a  =  d  :c. 

481.  Principle  6.  —  If  four  numbers  are  in  proportion,  the 
sum  of  the  terms  of  the  first  ratio  is  to  either  term  of  the  first  ratio 
as  the  sum  of  the  terms  of  the  second,  ratio  is  to  the  coi-responding 
term  of  the  second  ratio ;  that  is,  the  nmnbers  are  in  proportion 
by  composition. 

For,  given  a:b  =  c:d, 


or 


a  _  c 
b~d 


or 


Then,  ^  +  1  =  £  +  1, 

h  d 

a  -\-b      c  +  d. 


b      -     d     ' 
that  is,  a  +  b  :  b  =  c  +  d  :  d. 

Similarly,  taking  the  given  proportion  by  inversion  (Prin.  5),  and 
adding  1  to  both  members,  we  obtain 

a  -\-  b  :  a  =  c  -]-  d  :  c. 

482.  Principle  7.  —  If  four  numbers  are  in  proportion,  the 
difference  between  the  terms  of  the  first  ratio  is  to  either  terrn  of 
the  first  ratio  as  the  difference  betiveen  the  terms  of  the  second 
ratio  is  to  the  corresponding  term  of  the  second  ratio ;  that  is, 
the  numbers  are  i7i  proportion  by  division. 

For,  in  the  proof  of  Prin.  6,  if  1  is  .subtracted  instead  of  added,  the 
following  proportions  are  obtained  : 

a  —  b  :  b  =  e  —  d  :  d, 
and  a  —  b  :  a  =  c  —  d  :  c. 


RATIO  AND  PROPORTION  377 

483.  Principle  8.  —  If  four  numbers  are  in  proportiouj  the 
stun  of  the  terms  of  the  first  ratio  is  to  their  difference  as  the  sum 
of  the  terms  of  the  second  ratio  is  to  their  difference;  that  is,  tJie 
numbers  are  in  jjroportion  by  composition  and  division. 

For,  given  a.b  =  c:  d. 

Byl'rin.O,  £±_*  =  1±^.  '  (1) 

h  a 

•^  b  d  ^  ^ 


Dividing  (1)  by  (J),  Ax.  4, 
that  is,  a  -\-  b  :  a  —  h  =  c  ^  d  : 


a  +  b  _c  -\-  d  ^ 
a  —  b      c  —  d^ 


484.  Principle  9.  —  If  four  numbers  are  in  projwrtion,  their 
like  powers,  and  also  their  like  roots,  are  in  proportion. 

For,  given  a:b  =  c:d, 

b    d 

rheu,  Ax.  6  and  §  276,  3,  «!!=£!!; 

6"      d'* 

that  is,  a"  : //»  =  c"  :  r/". 

Also,  Ax.  7  and  §  291,  regarding  only  principal  roots, 

y/b      Vd'. 
that  is,  )^y/a  :  v^^  =  y/c  :  Vd. 

485 .  Princi  PLE  10. — In  a  proportion,  if  both  terms  of  a  couplet, 
or  both  antecedents,  or  both  consequents  are  mtdtiplied  or  divided 
1 1 'I  the  same  number,  the  restdting  four  numbers  form  a  proportion. 

For.  'jfivfTi  a:h  =  c:d, 

a     c 

ol-  -  =  -  . 

b     d 

Then,  §  1J>5.  ***^ 

mb     nd 

Also,  Ax.  8, 

b      n      d 


378  RATIO   AND   PROPORTION 

486.  Principle  11.  —  TJie  prodacts  of  correapondiny  terms  of 
any  number  of  proportions  form  a  proportion. 

For,  given  a  :  b  =  c  :  d, 

k :  I  =  m  :  n, 
and  X :  y  =  z  :  w. 

Writing  each  proportion  as  a  fractional  equation,  we  have 

a      c     k      m  ■,  X       z 

-  =  -,  -  —  —,  and  -  =  — . 
h      d     I       n  y      'w 

Multiplying  these  equations,  member  by  member.  Ax.  3,  we  have 
akx  _  cmz  . 
bly       dnw  ' 
that  is,  akx  :  hly  =  cmz  :  dnw. 

487.  Principle  12.  —  If  two  proportions  have  a  common 
couplet,  the  other  two  couplets  ivill  form  a  proportion. 

For,  given  a  :b  =  x  -.y, 

and  c  :d  =  x:y. 

Then,  Ax.  .5,  a  :  b  =  c  :  d. 

488.  A  proportion  that  consists  of  three  or  more  equal 
ratios  is  called  a  multiple  proportion. 

2:4  =  3:6  =  5: 10  and  a:b  =  c  :  d  =  e  :f  are  multiple  proportions. 

489.  Principle  13.  —  In  any  multiple  proportion  the  sum  of 
all  the  antecedents  is  to  the  sum  of  all  the  co7isequents  as  any 
antecedent  is  to  its  consequent. 


For,  given 

a  :b—  c  :d  =  e  :  f 

or 

«  =  :^  =  ^=r,thevah 
b      d     f 

Then,  Ax.  3, 

a  =  br,  c  =  dr,  e  =  fr ; 

whence,  Ax.  1, 

a  +  c 

+  e=  (b  +  d-^f)r, 

b-^d 

+  e      ^.       a      c  _  e  . 
+  /              b      d     7' 

RATIO   AND  PROPORTION  379 

490.  A  multiple  pioportipu  in  which  each  consequent  is 
repeated  as  the  antecedent  of  the  following  ratio  is  called  a 
continued  proportion. 

•J  :  4  =  4  :  b  =  8 :  10  ajul  a  :  b  =  b  :  c  =  c  :  d  are  continued  proportions. 

491.  Principle  14.  —  If  three  numhera  are  in  contmup<\  ))rn- 
jH)rtion,  the  ratio  of  the  extremes  is  equal  to  the  square  of  f  it  Jut 
giren  ratio. 


0) 
(-0 


For,  given 

a:b  =  b:c, 

or 

a_b 
b      c' 

Multiplying  by  ?,  Ax.  3, 
b 

a*     a 
b^'c' 

By  (1)  and  Prin.  9, 

a«      b^ 
b^      c*' 

By  (2)  and  (3),  Ax.  5, 

a      a2      b\ 

c      6«      c2 ' 

that  is, 

n:c  =  a^:b^  =  b-':c^. 

492.  J*iMN'Cii'LK  15.  —  If  four  numlters  are  in  continual  pm- 
portion,  the  ratio  of  the  extremes  is  equal  to  the  cube  of  any  of  the 
ffiren  ratios. 

For,  given  a:b  =  b:c  —  c:d^ 


a  __  o  _  c 
b'c'fi' 

(1) 

be       n      a      a  _  «*. 
c  '  d~b  '  b  '  b~  b"^' 

C-?) 

a:d  =  a»:b*. 

('0 

a*:b*=:b*:c*=c*:  </«. 

(4) 

,/  =  rt»  :  6«  =  i/«  :  c»  =  r«  :  r/«. 

or 

Then,  Ax.  5,  - 

til  at  is,  canceling. 
By  (1)  and  Prin.  0, 
By  (4)  and  (3),  Ax.  :.. 

EXERCISES 

493.    1.    Find  the  value  of  x  in  the  proportion  .3  :  5  =  a; :  55. 
Solution.  3  :  5  =  a; :  66. 

Prin.  2,  x=ll^=83. 


380  KA no   AND   PROPORTION 

Find  the  value  of  x  in  each  of  the  following  proportions : 

2.  2:3  =  6:0.-.  5.    0^  + 2  :  a^  =  10  :  6. 

3.  5  :  a;  =  4  :  3.  6.    a; :  x  —  1  =  15  :  12. 

4.  X'.x^X'.^.  7.    .T  +  2:a;-2  =  3:l. 

8.  Show  that  a  mean  proportional  between  any  two  num- 
bers having  like  signs  has  the  sign  ± . 

9.  Find  two  mean  proportionals  between  V2  and  V<S. 

10.  Find  a  third  proportional  to  4  and  6. 

11.  Find  a  fourth  proportional  to  3,  8,  and  1\. 

12.  Find  a  mean  proportional  between 

x^^x-^^  and  ^-t^^zl?. 
a;  +  4  x^'l 

Test  to  see  whether  the  following  are  true  proportions : 

13.  51:3  =  4:11  15.    5  :  7  a:  =  10  :  14a-. 

14.  4:13  =  2:61  16.    2.4a  :  .8  a  =  6a  :  2a. 

17.    Given  a  :  b  =  c  :  d, 

to  prove  that  2a  +  5&:4a-36  =  2c  +  5r?:4c-3d. 

Proof.  —  First  form,  from  the  given  proportion,  a  proportion  hav- 
ing as  antecedents  the  antecedents  of  the  required  proportion. 

a:h^c:d.  (1) 

Prin.  10,  2a'.h^2c:d. 

Priu.  10,  2  a  :  5  ?>  =  2  c  :  5  r7. 

Prin.6,  2a  +  oh:2a  =  2  c  +  'od  :2  c.  (2) 

Next  form  from  (1)  a  proportion  having  as  antecedents  the  conse- 
quents of  the  required  proportion. 

Prin.  10,  4  a  :  ?>  =  4  c  :  c?. 

Prin.  10,  4a:36  =  4c:3f/. 

Prin.  7,  4a  -  3  &  :  4rt  =:  4c  -  3rf  :  46-.  (3) 

Prin.  10,  4a-3&:2a  =4c'-3J:2c.  (4) 

Next  take  (2)  and  (4)  by  alternation  (Prin.  4)  and  apply  Prin.  12 
to  the  results. 

Then,  2  a  +  5  6  :  2  c  +  5  c?  =  4  a  -  3  ft  :  4  c  -  3  (/, 

or,  Prin.  4,  2a  +  56:4a-3&  =  2c-|-5J:4c-3^/. 


8. 

d.b  =  f:  a. 

9. 

c  :  d  =     :  - 
b   a 

RATIO  AND  PROPORTION  381 

Wlien  (I  :b  =  c:dy  prove  that  the  following  are  true  propor- 
tions : 

21.  a-:b-r=l:cP. 

22.  ma:-  =  mc:-' 
20.   6^ : d»  =  a" : c*.                         23.   ac'.bd  =  (^:d^, 

24.  Vad:V6  =  Vc:l. 

25.  a-|-/>:c--|-d  =  a  — ^:  c  — cZ. 

26.  a:a-\-b  =  a-\-c:a-\-b-\-c-\-(l. 

27.  rt  +  /^:rH-f/  =  V^T^:  VcH^. 

28.  fr*-|-a=^6-|-a/r  +  /'^:  a''  =  c-"'  +  c*dH-cd2  +  (f:(r\ 

29.  2n  +  'db'.'^<i-\-Ab  =  2c-{-^d:  .S  c  -f  4  d. 

30.  2  a  +  3  f :  2  a  -  3  c  =  8  /^  +  1 2  fZ :  8  /^  -  1 2  f /. 

31.  (( -f  6  +  r  -h  '^ :  <^  —  '-^  H-  '•  —  d  =  a  H-  6  —  0  —  d :  a  —  6  —  c  -f  d 

32.  If  rt :  6  =  r  :  f/,  and  if  x  be  a  third  proportional  to  a  and  6, 
and  //  a  third  proportional  to  b  and  c,  show  that  the  mean  pro- 
portional between  x  and  y  is  equal  to  that  between  c  and  d. 

33.  Solve  for  »,  V^^T7+_V^^  V^  +  7- Vi. 

4  H-  Va;  4  —  Vi 

Solution 
By  alternation,  Prin.  4, 

y/x-k-1  ■k-y/x_A-\-Vx 
y/x  -I-  7  —  Vx     4  —  Vx 
By  composition  and  division,  Prin.  8, 


2V3cT7^    8 
2Vx        2>/x 
Since  the  consequents  are  equal,  the  antecedents  are  equal. 


Therefore,  2  Vx  +  7  =  8. 

Solving,  x  =  9. 


382  RATIO   AND  PROPORTION 


34.    Given       — :==^ —  =  —       '        ^ — ^,  to  solve  for  x. 
Va;  +  ll-2      V2a;H-14-2| 


Solution 
By  composition  and  division,  Priri.  8, 


2>/y.+  11^2V2a;  +  14 
4  -L«- 

Dividing  both  terms  of  each  ratio  by  2,  Prin.  10, 

va;+  11  _  V2x+  ]i 
2        ~  I         * 

Dividing  the  consequents  by  |,  Prin.  10, 


Vx+11      V2  cc  4- 14 


Prin.  4, 

3 

_3 
4* 

4 

By  alternation, 

Va;  +  ll 

V2  X  +  14 

Squaring  and  applying  : 

Prin.  7, 

x  +  S 

7 

x  +  11 

9 

Solving, 

X: 

=  25. 

Solve  for  x  by  the  principles  of  proportion : 
35.     VS+V^_TO.  3g_ 


V  it'  —  V  m      ^* 

36.    V^+V2^^2  33 

^x-V2a     1 

37.  ^±y^_i3,      4Q    _        _ 

x—^x  —  1       '^  Vax-j-b     3Vax-\-ob 

.^      Va  +  Va  +  a;  _  V^  +  Vx  —  b 


V^  +  &  H-  Vic  - 

-b 

Vir  +  6  —  Vo.'  - 

^ 

Va  +  V«  —  .^* 

1 

a 

Va  —  Va  —  a; 

Vo^  —  b      S^ax  — 

-2b 

V«  —  Va  +  .'y      Vft  —  V;v  —  & 


>.«      VxH-l4-Va?  — 2      V«  — 3  +  Va.'  — 4 

42.     r= =  = ^:=z: === 

■y/x  +  1  —  Vit'  —  2       V't"  —  3  —  V^'  —  4 


RATIO   AND  PROPORTION  383 

Problems 

494.  1.  Divide  $35  between  two  men  so  that  their  shares 
shall  be  in  the  ratio  of  3  to  4. 

2.  Two  numbers  are  in  the  ratio  of  3  to  2.  If  each  is 
increased  by  4,  the  sums  will  be  in  the  ratio  of  4  to  3.  What 
are  the  numbers  ? 

3.  Divide  16  into  two  parts  such  that  their  product  is  to 
the  sum  of  their  squares  as  3  is  to  10. 

4.  Divide  25  into  two  parts  such  that  the  greater  increased 
by  1  is  to  the  less  decreased  by  1  as  4  is  to  1. 

5.  The  sum  of  two  numbers  is  4,  and  the  square  of  their 
sum  is  to  the  sum  of  their  squares  as  8  is  to  5.  What  are  the 
numbers  ? 

6.  Find  a  number  that  added  to  each  of  the  numbers  1,  2, 
I.  and  7  will  give  four  numbers  in  proportion. 

7.  In  the  state  of  Minnesota  the  ratio  of  native-born  in- 
liabitants  to  foreign-bom  recently  was  5:2.      What  was  the 

;mber  of  each,  if  the  total  population  was  1,750,000  ? 

8.  A  business  worth  $  19,000  is  owned  by  three  partners. 
Tlie  share  of  one  partner,  $6000,  is  a  mean  proportional  be- 
t  ween  the  shares  of  the  other  two.     Find  the  share  of  each. 

9.  What  number  must  be  added  to  each  of  the  numbers  11, 
1  7,  2,  and  5  so  that  the  sums  shall  be  in  proportion  when 
tiiken  in  the  order  given  ? 

10.  Four  numbers  are  in  proportion ;  the  difference  between 
ill*'  first  and  the  third  is  2j;  the  sum  of  the  second  and  the 
third  is  6J;  the  third  is  to  the  fourth  as  4:5.     Find  the  num- 

l>t*rs. 

11.  Prove  that  no  four  consecutive  integers,  as  ?i,  n -f  1, 
//  4-  2,  and  n  -f  3,  can  form  a  proportion. 

12.  Prove  that  the  ratio  of  an  odd  number  to  an  even  num- 
1,  as  2m  +  l:2?i,  cannot  be  equal  to  the  ratio  of  another 

t  \'en  number  to  another  odd  number,  as  2  a; :  2  ^  + 1« 


884 


RATIO    AND   PROPORTION 


13.    The  area  of  tlie  right  triangle  shown  in  Fig.  1  may  be 

expressed  either  as  I  ah  or  as  \  ch.     Form  a  proportion  whose 
terms  shall  be  a,  b,  c,  and  h. 


Fig.  1. 


Fig.  2. 


Fig.  3. 


14.  In  Fig.  2,  the  perpendicular  p,  which  is  20  feet  long,  is 
a  mean  proportional  between  a  and  b,  the  parts  of  the  diame- 
ter, which  is  50  feet  long.     Find  the  length  of  each  part. 

15.  In  Fig.  3,  the  tangent  t  is  sl  mean  proportional  between 
the  whole  secant  c-\-  e,  and  its  external  part  e.  Find  the 
length  of  ^,  if  e  =  9f  and  c  =  50f . 

16.  The  strings  of  a  musical  instrument  produce  sound  by 
vibrating.  The  relation  between  the  number  of  vibrations 
N  and  N'  of  two  strings,  different  only  in  their  lengths  I  and  /', 
is  expressed  by  the  proportion 

A  c  string  and  a  g  string,  exactly  alike  except  in  length, 
vibrate  256  and  384  times  per  second,  respectively.  If  the  c 
string  is  42  inches  long,  find  the  length  of  the  g  string. 

17.  If  L  and  I  are  the  lengths  of  two  pendulums  and  T  and  t 
the  times  they  take  for  an  oscillation,  then 

T':f  =  L:l. 

A  pendulum  that  makes  one  oscillation  per  second  is  approxi- 
mately 39.1  inches  long.  How  often  does  a  pendulum  156.4 
inches  long  oscillate  ? 

18.  Using  the  proportion  of  exercise  17,  find  how  many  feet 
long  a  pendulum  would  have  to  be  to  oscillate  once  a  minute. 


VARIATION 


495.  Many  problems  and  discussions  in  mathematics  have 
to  do  with  numbers  some  of  which  have  values  that  are  con- 
tinually dianging  while  others  remain  the  same  throughout  the 
discussion.  Numbers  of  the  first  kind  are  called  variables; 
numbers  of  the  second  kind  are  called  constants. 

Thus,  the  distance  of  a  moving  train  from  a  certain  station  is  a  varia- 
.  but  the  distance  from  one  station  to  another  is  a  constant. 

Two  variables  may  be  so  related  that  when  one  changes  the 
other  changes  correspondingly. 

496.  One  quantity  or  number  is  said  to  vary  directly  as 
another,  or  simply  to  vary  as  another,  when  the  two  depend 
upon  each  other  in  such  a  manner  that  if  one  is  changed  the 
other  is  changed  in  the  same  ratio. 

I'hus,  if  a  man  earns  a  certain  sum  per  day,  the  amount  of  wages  he 
.:ns  varies  as  the  number  of  days  he  works. 

497.  The  sign  of  variation  is  oc.     It  is  read  ^varies  as.^ 
Thus,  X  X  y,  read  '  x  varies  as  y ',  is  a  brief  way  of  writing  the  proportion 

Z:x'  =  y:y', 
in  which  x'  is  the  value  to  which  x  is  changed  when  y  is  changed  to  y'. 

498.  The  expression  xacy  means  that  if  x  is  doubled,  y  is 
doubled,  or  if  x  is  divided  by  a  number,  y  is  divided  by  the 
same  number,  etc. ;  that  is,  that  the  ratio  of  a;  to  y  is  always 
the  same,  ov  constant. 

If  the  constant  ratio  is  represented  by  A;,  then  when  xxy, 

-  =  A:,  or  X  =  ky.     Hence, 

Ifx  varies  asy,  x  w  equal  to  y  multiplied  by  a  constant. 
3iilxk's  stand,  alo. — 25         385 


886  VARIATION 

499.  One  quantity  or  number  varies  inversely  as  another 
when  it  varies  as  the  reciprocal  of  the  other. 

Thus,  the  time  required  to  do  a  certain  piece  of  work  varies  inversely 
as  the  number  of  men  employed.  For,  if  it  takes  10  men  4  days  to  do  a 
piece  of  work,  it  will  take  5  men  8  days,  or  1  man  40  days,  to  do  it. 

In  a;  oc  - ,  if  the  constant  ratio  of  x  to  -  is  k.  -  =  k,orxv  =  k. 
y  ?/        '  1       '        ^ 

Hence,  y 

If  X  varies  inversely  as  y,  their  product  is  a  constant. 

500.  One  quantity  or  number  varies  jointly  as  two  others 
when  it  varies  as  their  product. 

Thus,  the  amount  of  money  a  man  earns  varies  jointly  as  the  number 
of  days  he  works  and  the  sum  he  receives  per  day.  For,  if  he  should  work 
three  times  as  many  days,  and  receive  twice  as  many  dollars  per  day,  he 
would  receive  six  times  as  much  money. 

In  a;  oc  yz,  if  the  constant  ratio  of  x  to  yz  is  k, 

—  =  k,  OT  X  =  kyz.     Hence, 

yz 

If  X  varies  jointly  as  y  and  z,  x  is  equal  to  their  product  multi- 
plied by  a  constant. 

501.  One  quantity  or  number  varies  directly  as  a  second  and 
inversely  as  a  third  when  it  varies  jointly  as  the  second  and  the 
reciprocal  of  the  third. 

Thus,  the  time  required  to  dig  a  ditch  varies  directly  as  the  length  of 
the  ditch  and  inversely  as  the  number  of  men  employed.  For,  if  the  ditch 
were  10  times  as  long  and  5  times  as  many  men  were  employed,  it  would 
take  twice  as  long  to  dig  it. 

1  y 

In  X  cc  y  '  -,  OY  X  cc  -,  if  A;  is  the  constant  ratio, 

z  z 

x-i--  =  k,  or  x  =  k-.     Hence, 

z  z 

Ifx  xaries  directly  as  y  and  inversely  as  z,  x  is  equal  to  -  midti- 
plied  by  a  constant. 


VARIATION  887 

502.  If  X  varies  as  //  when  z  is  coiis/an/.  <i,i(l  x  varies  oa  z 
when  y  is  constant y  then  x  varies  as  >i\  f/nn  both  y  and  z  are 
variable. 

Thus,  the  area  of  a  triangle  varies  as  the  base  when  the  altitude  is  con- 
stant ;  as  the  altitude  when  the  base  is  constant ;  and  as  the  product  of 
the  ba.se  and  altitude  when  l>oth  vary. 

Proof 

Since  the  variation  of  x  depends  upon  the  variations  of  y  and  2,  sup- 
pose the  latter  variations  to  take  place  in  succession,  each  in  turn  pro- 
ducing a  corresponding  variation  in  x. 

While  z  remains  constant,  let  y  change  to  ;/,,  thus  causing  x  to 
.  li:inge  to  x'. 

Then,  §  =  ^.  (1) 

x'     yx 

Now  while  y  keeps  the  value  y,,  let  2  change  to  2,,  thus  causing  x' 

tn  change  to  x,. 

Then,  £!  =  £.  (2) 

X,      2, 

Multiplying  (1)  by  (2),        ^  =  J^ .  (:^) 

x  =  -^.y2.  (4) 

Since,  if  both  changes  are  made,  Xp  y,,  aii<l  :,  air  constants,  —J-  is  a 
constant,  which  may  be  represented  by  k.  ^*  ^ 

TIkmi,  (4)  becomes  x  =  kyz. 

Hence,  X  cc  yz. 

Similarly,  if  x  varies  as  each  of  three  or  more  numbers,  y,  z, 
r,  -"  when  the  others  are  constant,  when  all  vary  x  varies  as 
their  product. 

That  is,  xccyzv-'. 

Thus,  the  volume  of  a  rectangular  solid  varies  as  tlie  length,  if  the  width 
and  thickness  are  constant ;  as  the  width,  if  the  length  and  thickness  are 
constant ;  as  the  thickness,  if  the  length  and  width  are  constant ;  as  the 
product  of  any  two  dimensions,  if  the  other  dimension  is  constant;  and 
as  the  product  of  the  three  dimensions,  if  all  \ aiy. 


388  VARIATION 


EXERCISES 

503.  1.  If  a;  varies  inversely  as  y^  and  a;  =  6  when  ^  =  8, 
what  is  the  value  of  x  when  y  =  12? 

Solution 
•   Since  aj «  -,  let  A:  be  the  constant  ratio  of  a;  to  - • 

y  y 

Then,  §  499,  xy  =  k.  (1) 

Hence,  when  x  —  Q  and  y  =  8,    ^  =  6  x  8,  or  48.  (2) 

Since  ^  is  constant.  A;  =  48  when  y  =  12, 

and   (1)  becomes  12  a:  =  48. 

Therefore,  when  ?/  =  12,  x  =  A. 

2.  If  ic  oc "-,  and  if  a?  =  2  when  y  =  12  and  z  =  2,  what  is  the 

z 

value  of  X  when  y  =  M  and  z  =  l? 

3.  If  icx-,  and  if  x  =  60  when  2/  =  24  and  z  =  2,  what  is 
the  value  of  y  when  a;  =  20  and  z  =  1? 

4.  If  a;  varies  jointly  as  y  and  z  and  inversely  as  the  square 
of  w,  and  if  a;  =  30  when  ?/  =  3,  2;  =  5,  and  z^  =  4,  what  is  the    , 
value  of  X  expressed  in  terms  of  y,  z,  and  w  ? 

5.  If  a;  Qc  2/ and  2/ X  2,  prove  that  a;  oc  2;. 

Proof 

Since  x  ccy  and  y  ccz,  let  m  represent  the  constant  ratio  of  x  to  y, 
and  m  the  constant  ratio  of  y  to  z. 

Then,  §498,  x  =  my,  (1) 

and  y  =  nz.  (2) 

Substituting  nz  for  «/  in  (1),      a:  =  mn2.  (3) 

Hence,  since  mn  is  constant,       x  ccz. 


6.  If  a;x-,  and  ?yoc  -,  prove  that  a;oc2. 

2/  '       2; 

7.  li  xccy  and  «  x?/,  prove  that  (a;  ±  z)  ccy. 


VARIATION  889 

Problems 

504.  1.  The  volume  of  a  cone  varies  jointly  as  its  altitude 
and  the  square  of  the  diameter  of  its  base.  When  the  altitude 
is  15  and  the  diameter  of  the  base  is  10,  the  volume  is  392.7. 
What  is  the  volume  when  the  altitude  is  5  and  the  diameter 
of  the  base  is  20  ? 

Solution 

I^t  F,  £r,  and  D  denote  the  volume,  altitude,  and  diameter  of  the  base, 
respectively,  of  any  cone,  and  V  the  volume  of  a  cone  whose  altitude  is  6 
and  the  diameter  of  whose  base  is  20. 

Since  V  x  IIDS  or  F  =  kHD\ 

and  F  =  392.7  when  if  =  15  and  D  =  10, 

392.7  =  ifc  X  16  X  100.  (1) 

Also,  since  V  becomes  V'  when      H—b  and  D  =  20, 

F'  =  ifc  X  6  X  400.  (2) 

Dividing  (2)  by  (1),  Ax.  4,       -^  =  ^  ^  ^^  - 1  (3) 

^  ^  ^    "^  ^  ^'  392.7      16  X  100     3  ^  ^ 

:.  F'  =  J  of  392.7  =  623.6. 

2.  The  circumference  of  a  circle  varies  as  its  diameter.  If 
the  circumference  of  a  circle  whose  diameter  is  1  foot  is  3.1416 
feet,  find  the  circumference  of  a  circle  100  feet  in  diameter. 

3.  The  area  of  a  circle  varies  as  the  square  of  its  diameter. 

I  f  the  area  of  a  circle  whose  diameter  is  10  feet  is  78.54  square 
teet,  what  is  the  area  of  a  circle  whose  diameter  is  20  feet  ? 

4.  The  distance  a  body  falls  from  rest  varies  as  the  square 
of  the  time  of  falling.  If  a  stone  falls  64.32  feet  in  2  seconds, 
liow  far  will  it  fall  in  5  seconds? 

5.  The  volume  of  a  sphere  varies  as  the  cul^e  of  its  diameter. 

I I  the  ratio  of  the  sun's  diameter  to  the  earth's  is  109.3,  how 
many  times  the  volume  of  the  earth  is  the  volume  of  the  sun  ? 

6.  If  10  men  can  do  a  piece  of  work  in  20  days,  how  long 
will  it  take  25  men  to  do  it  ? 

7.  If  a  men  ran  do  a  piece  of  work  in  h  days,  how  many 
men  will  be  required  to  do  it  in  c  days  ? 


390  VARIATION 

8.  The  area  of  a  triangle  varies  jointly  as  its  base  and 
altitude.  The  area  of  a  triangle  whose  base  is  12  inches  and 
whose  altitude  is  6  inches  is  36  square  inches.  What  is  the 
area  of  a  triangle  whose  base  is  8  inches  and  whose  altitude  is 
10  inches  ?     What  is  the  constant  ratio  ? 

9.  A  wrought-iron  bar  1  square  inch  in  cross  section  and 
1  yard  long  weighs  10  pounds.  If  the  weight  of  a  uniform  bar 
of  given  material  varies  jointly  as  its  length  and  the  area  of 
its  cross  section,  what  is  the  weight  of  a  wrought-iron  bar  36 
feet  long,  4  inches  wide,  and  4  inches  thick  ? 

10.  The  weight  of  a  beam  varies  jointly  as  the  length,  the 
area  of  the  cross  section,  and  the  material  of  which  it  is  com- 
posed. If  wood  is  y^2  ^s  heavy  as  wrought  iron  (see  exercise  9), 
what  is  the  weight  of  a  wooden  beam  24  feet  long,  12  inches 
wide,  and  12  inches  thick  ? 

11.  What  is  the  weight  of  a  brick  2  in.  x  4  in.  x  8  in.,  if 
the  material  is  ^  as  heavy  as  wrought  iron?  (For  the  weight 
of  wrought  iron,  see  exercise  9.) 

12.  The  distances,  from  the  fulcrum  of  a  lever,  of  two 
weights  that  balance  each  other  vary  inversely  as  the  weights. 
If  two  boys  weighing  80  pounds  and  90  pounds,  respectively, 
are  balanced  on  the  ends  of  a  board  8^  feet  long,  how  much  of 
the  board  has  each  on  his  side  of  the  fulcrum  ? 

13.  A  water  carrier  carries  two  buckets  of  water  suspended 
from  the  ends  of  a  4-foot  stick  that  rests  on  his  shoulder.  If 
one  bucket  weighs  60  pounds  and  the  other  100  pounds,  and 
they  balance  each  other,  what  point  of  the  stick  rests  on  his 
shoulder  ? 

14.  The  horse  power  {H)  that  a  solid  shaft  can  transmit 
safely  varies  jointly  as  its  speed  in  revolutions  per  minute  {N) 
and  the  cube  of  its  diameter.  A  5-inch  solid  steel  shaft  making 
150  revolutions  per  minute  can  transmit  585  horse  power.  How 
many  horse  power  could  it  transmit  at  half  this  speed,  if  its 
diameter  were  increased  1  inch  ? 


VARIATION  391 

15.  The  weight  of  a  bcxly  near  the  earth  varies  inversely  as 
tiie  square  of  its  distance  from  the  center  of  the  earth.  If  the 
radius  of  the  earth  is  4000  miles,  what  would  be  the  weight  of 
;i  i-pound  brick  4000  miles  above  the  earth's  surface? 

16.  The  weight  of  wire  of  given  material  varies  jointly  as 
tlie  length  and  the  square  of  the  diameter.     If  3  miles  of  wire 

•S  of  an  inch  in  diameter  weighs  288  pounds,  find  the  weight 
"f  V  mile  of  wire  .16  of  an  inch  in  diameter. 

17.  The  illumination  from  a  source  of  light  varies  inversely 
IS  the  square  of  the  distance.     How  far  must  a  screen  that  is 

10  feet  from  a  lantern  be  moved  so  as  to  receive  one  fourth  as 
much  light? 

18.  The  number  of  times  a  pendulum  oscillates  in  a  given 
time  varies  inversely  as  the  square  root  of  its  length.  If  a 
pendulum  39.1  inches  long  oscillates  once  a  second,  what  is  the 
Itiigth  of  a  pendulum  that  oscillates  twice  a  second?  once  in 
three  seconds? 

19.  Three  spheres  of  lead  whose  radii  are  6  inches,  8  inches, 
and  10  inches,  respectively,  are  united  into  one.  Find  the 
radius  of  the  resulting  sphere,  if  the  volume  of  a  sphere  varies 
as  the  cube  of  its  radius. 

20.  The  volume  of  a  cone  varies  jointly  as  its  altitude  and 
tlie  square  of  the  diameter  of  its  base.  The  altitudes  of  three 
cones,  S,  P,  and  /?,  are  30  ft.,  10  ft.,  and  5  ft.,  respectively. 
The  diameter  of  the  base  of  P  is  5  ft.  and  that  of  R  is  10  ft. 
If  the  volume  of  S  is  equivalent  to  that  of  P  and  R  combined, 
what  is  the  diameter  of  the  base  of  S  ? 

21.  A  boy  wishes  to  ascertain  the  height  of  a  tower.  He 
knows  that  it  is  31  feet  6  inches  from  his  window  to  the  pave- 
ment below,  and  that  the  distance  through  which  a  body  falls 
varies  as  the  square  of  the  time  of  falling.  He  drops  a  marble 
from  his  window  and  finds  that  it  strikes  the  pavement  in  1.4 
seconds.  Then  throwing  a  stone  upward  he  observes  that  it 
takes  just  3  seconds  for  it  to  descend  from  the  top  of  the 
tower  to  the  ground.     What  is  the  height  of  the  tower? 


392  VARIATION- 

SOS.    Algebraic  expression  of  physical  laws. 

If  two  wires  exactly  alike  in  all  respects  except  in  length  (I) 
are  stretched  by  equal  weights,  the  greater  number  (n)  of  vibra- 


tions per  second  will  be  made  by  the  shorter  wire.    If  one  wire 
is  half  as  long  as  the  other,  its  rate  of  vibration  will  be  twice 
as  great;  if ^  as  long,  the  rate  will  be  3  times  as  great;  etc. 
This  result  is  expressed  by  the  variation 

1 


n  cc 


I 

Next,  experimenting  with  two  wires  alike   in  all  respects 

except  in  diameter  (d),  it  is  found  that 

1 
ncc  -. 
d 

Next,  excluding  all  variable    elements    in  the   experiment 
except  the  stretching  weight  (  TF),  it  is  found  that 

woe  VW^ 

Finally,  experimenting  with  w4res  of  different  materials,  as 
steel  and  brass,  which  have  different  specific  gravities  (s), 

1 


n  oc 


v; 


Since  the  number  of  vibrations  per  second  varies  inversely 
as  I  and  d,  directly  as  the  square  root  of  W,  and  inversely  as 
the  square  root  of  s,  by  §  502, 

Vw 


n  Gc 


IdVs 


which  is  the  expression  of  the  law  as  a  variation. 


VARIATION  393 

It  is  found  by  measuring  n,  /,  d,  W,  and  s  in  any  case  that 

the  constant  ratio  of  the  first  member  to  the  second  is  -d-  • 
Hence,  the  law  may  be  expressed  by  the  equation  ' 

Vw 


71  =  J-   . 

^  IT 


IdVs 


EXERCISES 

506.  In  the  following  use  3.1416  for  the  value  of  tt,  and 
32.10  or  980  for  the  numerical  value  of  g  according  as  the 
distance  unit  is  1  foot  or  1  centimeter.     Regard  g  as  constant. 

Express  by  a  variation,  and  when  k  is  given  by  an  equation, 
each  of  the  following  laws : 

1.  The  distance  (s)  passed  through  in  I  seconds  by  a  body 
falling  freely  from  a  state  of  rest  varies  as  the  square  of  the 
time.     The  constant  ratio  (A:)  is  equal  to  ^  g. 

2.  The  time  required  by  a  simple  pendulum  to  make  a 
complete  oscillation  varies  as  the  square  root  of  its  length. 
A:  =  2  TT  -^  Vgr  is  the  constant  ratio,  at  any  given  place. 

3.  The  velocity  (v)  acquired  by  a  body  falling  from  a  height 
{h)  varies  as  the  square  root  of  the  height.  The  constant  ratio, 
for  any  given  place,  is  V2  g. 

4.  The  quantity  (Q)  of  water  flowing  from  a  circular  orifice, 
of  diameter  (cl)  and  under  a  height,  or  head  (h),  of  water  varies 
;is  the  square  of  d  and  as  the  square  root  of  h.  The  constant 
ratio,  under  ordinary  conditions,  is  A;  =  .625  •  \  tr  V2  g. 

5.  The  intensity  of  a  current  (/)  in  an  electric  circuit  varies 
directly  as  the  electromotive  force  {E)  and  inversely  as  the  re- 
sistance {R)  in  the  circuit.     The  constant  ratio  is  1. 

6.  The  heat  loss  (P)  in  an  electric  circuit  varies  directly  as 
the  intensity  of  the  current  (/)  and  the  square  of  the  resist- 
ance (/?).     The  constant  ratio  is  1. 


PROGRESSIONS 


507.  A  succession  of  numbers,  each  of  which  after  the  first 
is  derived  from  the  preceding  number  or  numbers  according 
to  some  fixed  iaw,  is  called  a  series. 

The  successive  numbers  are  called  the  terms  of  the  series. 
The  first  and  last  terms  are  called  the  extremes,  and  all  the 
others,  the  means. 

In  the  series  2,  4,  6,  8,  10,  12,  14,  each  term  after  the  first 
is  greater  by  2  than  the  preceding  term.  This  is  the  law  of 
the  series.  Also  since  1st  term  =  2  •  1,  2d  term  =  2-2,  3d  term 
=  2-3,  etc.,  the  law  of  the  series  may  be  expressed  thus : 

nth  term  =  2  n. 

In  the  series  2,  4,  8,  16,  32,  64,  128,  each  term  after  the  first 
is  twice  the  preceding  term ;  or  expressing  the  law  of  the 
series  by  an  equation,  or  formula, 

nth.  term  =  2\ 

ARITHMETICAL  PROGRESSIONS 

508.  A  series,  each  term  of  which  after  the  first  is  derived 
from  the  preceding  by  the  addition  of  a  constant  number,  is 
called  an  arithmetical  series,  or  an  arithmetical  progression. 

The  number  that  is  added  to  any  term  to  produce  the  next  is 
called  the  common  difference. 

2,  4,  6,  8,  •••  and  15,  12,  9,  6,  •••  are  arithmetical  progressions.  In  the 
first,  the  common  difference  is  2  and  the  series  is  ascending  ;  in  the  sec- 
ond, the  common  difference  is  —  3  and  the  series  is  descending. 

A.  P.  is  an  abbreviation  of  the  words  arithmetical  progression. 

394 


^  PROGRESSIONS  395 

509.  To  find  the  nth,  or  last,  term  of  an  arithmetical  series. 
Ill  the  arithmetical  series 

1,  3,  5,  7,  9,  11,  13, 15,  17, 19, 

the  com^mon  difference  is  2,  or  d=  2.  This  difference  enters 
f))ice  in  the  second  term,  for  3  =  1  -f  (/ ;  twice  in  the  thii'd  term, 
{ ur  5  =  1  -|-  2  d ;  three  times  in  the  fourth  term,  for  7  =  1  -f  3  cZ ; 
;iiid  so  on  to  the  10th,  or  last,  term,  which  equals  1  -|-9  d. 

In  ((,  a-\-d,  a-\-2dy  a-\-3d,  ••-, 

which  is  the  general  form  of   an  arithmetical  progression,  a 
representing  the   hrst   term    and   d   the   common    difference, 
observe  that  the   coefficient  of  d  in  the   expression   for  any 
term  is  one  less  tlian  the  number  of  the  term. 
Then,  if  the  nth,  or  last,  term  is  represented  by  I, 

l  =  a  +  (n-  l)d,  (I) 

Note.  —  The  common  difference  d  may  be  either  positive  or  negative. 
Ill  the  A.  P.   25,  23,  21,  19,  17,  15,  d  =  -  2. 

EXERCISES 

510.  1.    What  is  the  10th  term  of  the  series  6,  9,  12,  •  •  ? 

PROCESS  Explanation.  —  Since  the  series  6,  9,  12,  •••  is 

,_  an  A. P.  the  common  difference  of  whose  terms  is 

^  _  a  4-  (7i  —  l)cf  g^  ^y  substituting  6  for  a,  10  for  n,  and  3  for  d  in 

—  ^-^  (10  —  1)3  the  formula  for  the  last  terra,  the  last  term  is  found 

=  33  to  be  33. 

2.  Find  the  20th  term  of  the  series  7,  11,  15,  ••.. 

3.  Find  the  16th  term  of  the  series  2,  7,  12,  •••. 

4.  Find  the  24th  term  of  the  series  1,  16,  31,  •••. 

5.  Find  the  18th  term  of  the  series  1,  8,  15,  •••. 

6.  Find  the  13th  term  of  the  series  —  3,  1,  5,  •  • . 

7.  Find  the  49th  term  of  the  series  1,  IJ,  1|-". 


396  PROGRESSIONS 

8.  Find  the  15th  term  of  the  series  45,  43,  41,  •-.. 
Suggestion.  — The  common  difference  is  —  2. 

9.  Find  the  10th  term  of  the  series  5,  1,  —  3,  •••. 

10.  Find  the  16th  term  of  the  series  a,  3  a,  5  a,  •••. 

11.  Find  the  7th  term  of  the  series  x  —  Sy,  x  —  2y,  ••-. 

12.  A  body  falls  16^2  feet  the  first  second,  3  times  as  far 
the  second  second,  5  times  as  far  the  third  second,  etc.  How- 
far  will  it  fall  during  the  10th  second  ? 

511.  To  find  the  sum  of  n  terms  of  an  arithmetical  series. 

Let  a  represent  the  first  term  of  an  A.P.,  d  the  common  dif- 
ference, I  the  last  term,  n  the  number  of  terms,  and  s  the  sum 
of  the  terms. 

Write  the  sum  of  n  terms  in  the  usual  order  and  then  in  the 
reverse  order,  and  add  the  two  equal  series ;  thus,  k 

s  =  a  +  (a  +  d)  4-  (a  +  2  d)  +  (a  +  3  d)  H \-l 

s=Z  +  (/-d)+  (Z-2d)+(Z-3d)+  ...  4-a. 

2  s  =  (a +  /)  +  («+ 0  +  («  +  0  +  («  +  0+ •••  +  («  + 0- 
.'.2s  =  n{a  +  l). 

s  =  ^(a-\-l),ovn(^^-  .       (11) 

EXERCISES 

512.  1.    Find  the  sum  of  20  terms  of  the  series  2,  5,  8,  •••. 

PROCESS 

Z  =  a+(n-l)d  =  2+(20-l)x3  =  59 
.  =  J^^  =  20/^5^V610 


Explanation.  —  Since  the  last  term  is  not  given,  it  is  found  by  for- 
mula I  and  substituted  for  I  in  the  formula  for  the  sum. 


PROGRESSIONS  397 

Find  the  sum  of : 

2.  16  terras  of  the  series  1,  5,  9,  •••. 

3.  10  terms  of  the  series  —  2,  0,  2,  •••. 

4.  6  terms  of  the  series  1,  3J,  6,  •••. 

5.  8  terms  of  the  series  a,  3  a,  5  a,  •••. 

6.  n  terms  of  the  series  1,  7,  13,  •••. 

7.  a  terms  of  the  series  x,  x-\'2 a,  •••. 

8.  7  terms  of  the  series  4,  11,  18,  •••• 

9.  10  terms  of  the  series  1,  —  1,  —  3,  •••. 

10.  10  terms  of  the  series  1,  |,  0,  •••. 

11.  How  many  strokes  does  a  common  clock,  striking  hours, 
make  in  12  hours  ? 

12.  A  body  falls  16^  feet  the  first  second,  3  times  as  far 
tlie  second  second,  5  times  as  far  the  third  second,  etc.  How 
far  will  it  fall  in  10  seconds  ? 

13.  Thirty  flower  pots  are  arranged  in  a  straight  line  4  feet 
apart.  How  far  must  a  lady  walk  who,  after  watering  each 
plant,  returns  to  a  well  4  feet  from  the  first  plant  and  in  line 
with  the  plants,  if  we  assume  that  she  starts  at  the  well  ? 

14.  How  long  is  a  toboggan  slide,  if  it  takes  12  seconds  for 
a  toboggan  to  reach  the  bottom  by  going  4  feet  the  first  second 
and  increasing  its  velocity  2  feet  each  second  ? 

15.  Starting  from  rest,  a  train  went  .18  feet  the  first  second, 
.."ri  feet  the  next  second,  .90  feet  the  third  second,  and  so  on, 
reaching  its  highest  speed  in  3  minutes  40  seconds.  How  far 
did  the  train  go  before  reaching  top  speed  ? 

16.  In  a  potato  race  each  contestant  has  to  start  from  a 
mark  and  bring  back,  one  at  a  time,  8  potatoes,  the  first  of 
which  is  6  feet  from  the  mark  and  each  of  the  others  6  feet 
farther  than  the  preceding.  How  far  must  each  contestant 
go  in  order  to  finish  the  race  ? 


398  PROGRESSIONS 

513.  The  two  fundamental  formulae, 

(I)  l  =  a+  {n - l)d  and  (II)  s  =  '^{a-{- 1), 

contain  j^ve  elements,  a,  d,  I,  n,  and  s.  Since  these  formulse  are 
independent  simultaneous  equations,  if  they  contain  but  two 
unknown  elements  they  may  be  solved.  Hence,  if  any  three  of 
the  live  elements  are  known,  the  other  two  may  be  found. 

EXERCISES 

514.  1.    Given  d  =  3,  Z  =  58,  s  =  260,  to  find  a  and  n. 

Solution 
Substituting  the  known  values  in  (I)  and  (II),  we  have 

58  =  «  +  (w  -  1).3,  or  «  +  3  w  =  61  ;  (1) 

and  260  =  \  n{a  +  58),  or  an  +  58  w  =  520.  (2) 

Solving,  n  =  i§4  or  5, 

and,  rejecting  n  =  ^^,  a  =  46. 

Since  the  number  of  terms  must  be  a  positive  integer,  fractional  or 
negative  values  of  n  are  rejected  whenever  they  occur. 

2.  Given  a  =  11,  r?  =  —  2,  s  =  27,  to  find  the  series. 

Solution 
Substituting  the  known  values  in  (I)  and  (II),  we  have 

/  =  11  +  (,i_l)(_2),  or  Z=13-2w;  (1) 

and  27  =  1  ;i(ll  +  0,  or  54  =  11  7i  +  In.  (2) 

Solving,  n  =  3  or  9  and  1  =  7  or  —  5.  (3) 

Hence,  the  series  is  11,  9,  7, 
or  11,  9,  7,  5,  3,  l,  _  l,  _  3,  -  5. 

3.  How  many  terms  are  there  in  the  series  2,  6,  10,  •••,  66  ? 

4.  What  is  the  sum  of  the  series  1,  6,  11,  •■•,  61  ? 


PKOGUESSIOXS  399 

5.  How  many  terms  are  there  in  the  series  —1,  2,  5,  •••,  if 

tlu;  Slim  i>  221  ? 

6.  Complete  the  series  2,  9, 10,  •••,  80. 

7.  Complete  the  series  —10,  —8^,  —7,  •••  to  10  terms. 

8.  The  sum  of  the  series--,  22,  27,  32,  ...  is  714.     If  there 
.11'  17  terms,  what  are  the  first  and  last  terms? 

9.  If  s  =  113f,  a  =  i,  andd  =  2,  find  w. 

10.  What  is  the  sura  of  the  series  —  10,  — 11,  —  0,    ••,  34  ? 

11.  What  is  the  sum  of  the  series    ••,  —  1,  3,  7,  •  •,  23,  if  the 
number  of  terms  is  16? 

12.  What  are  the  extremes  of  the  series  •••,8,  10,12,    ••,  if 
.s=  300,  and  n  =  20? 

13.  Find  an  A.  P.  of  14  terms  having  10  for  its  6th  term,  0 
for  its  11th  term,  and  98  for  the  sum  of  the  terms. 

14.  Find  an  A. P.  of  15  terms  such  that  the  sum  of  the  5th, 
♦ith,  and  7th  terms  is  60,  and  that  of  the  last  three  terms,  132. 

From  (I)  and  (II)  derive  the  formula  for : 

15.  /  in  terras  of  a,  n,  «.  18.    rJ  in  terms  of  a,  n,  s. 

16.  s  in  terras  of  a,  d,  I.  19.    d  in  terms  of  /,  n,  s. 

17.  a  in  terras  of  d,  n,  s.  20.    u  in  terms  of  a,  /,  8. 

515.    To  insert  arithmetical  means. 

EXERCISES 

1.    Insert  5  arithmetical  means  between  1  and  31. 
Solution 

Since  there  are  6  means,  there  must  be  7  terms.     Hence,  in  I  =  a  -\- 
(n  -  l)d,  Z  =  31,  a  =  1,  w  =  7,  and  d  is  unknown. 

Solving:,  d  =  5. 

Hence.  1,  fi.  11,  16,  21,  26,  31  is  the  series. 


400  PROGRESSIONS 

2.  Insert  9  arithmetical  means  between  1  and  6. 

3.  Insert  10  arithmetical  means  between  24  and  2. 

4.  Insert  7  arithmetical  means  between  10  and  —  14. 

5.  Insert  6  arithmetical  means  between  —  1  and  2. 

6.  Insert  14  arithmetical  means  between  15  and  20. 

7.  Insert  3  arithmetical  means  between  a  —  h  and  a  +  h. 

516.  If  A  is  the  arithmetical  mean  between  a  and  h  in  the 

series  .   , 

a,  A,  h, 

by  §508,  A-a  =  h-A. 

.',A  =  ^^.     That  is, 

Principle.  —  The  arithmetical  mean  hetiveen  two  numbers  is 
equal  to  half  their  su7n. 

EXERCISES 

517.  Find  the  arithmetical  mean  between : 

1-    fandj.  ^     x-\-y  ^^^  x-y 

1  7  X—  y  x4-y 

2.  a-\-b  and  a—b.  ^  ^ 

(1  —  xY 

3.  (a  -f-  by  and  (a  —  by.  1  +  ic 

Problems 

518.  Problems  in  arithmetical    progression   involving  two 
unknown  elements  commonly  suggest  series  of  the  form, 

X,  X  +  y,  x-\-2y,x-\-Sy,  etc. 

Frequently,  however,    the    solution    of   problems    is    more 
readily  accomplished  by  representing  the  series  as  follows: 

1.  When  there  are  three  terms,  the  series  may  be  written, 

x-y,x,x-\-y. 

2.  When  there  are  Jive  terms,  the  series  may  be  written, 

x-2y,x-y,x,x+y,x-h2y. 


PROGRESSIONS  401 

3.   When  there  are /our  terms,  the  series  may  be  written, 
X  -  3  y,  x  —  y,  X  +  y,  a;  +  3  y. 

The  sum  of  the  terms  of  a  series  represented  as  above  evidently  con- 

I  liiis  but  one  unknown  number. 

1.  The  sum  of  three  numbers  in  arithmetical  progression 

1-   30  and  the  sum  of  their  squares  is  462.     What  are  the 

numbers? 

Solution 

Let  the  series  be  x  —  y,  x,  x  +  y. 

Then,  (x  -  y)  +  x  +  (x  +  y)  =  30,  (1) 

a  ud  (X  -  y)2  +  x-i  +  (X  +  y)2  =  462.  (2) 

From  (1),  3x  =  30;  (3) 

whence,  x  —  10.  (4) 

From  (2),  3  x*  +  2  y-^  =  462.  (6) 

Substituting  (4)  in  (6),  2  \p-  =  162. 

Solving,  y  =  ±  9. 

Forming  the  series  from  x  =  10  and  y  =  ±  9,  the  terms  are 
1,  10,  19  or  19.  10,  1. 

2.  The  sum  of  three  numbers  in  arithmetical  progression  is 
IS.  and  tlieir  product  is  120.     What  are  the  numbers  ? 

3.  The  sum  of  three  numbers  in  arithmetical  progression  is 
- 1 ,  and  the  sum  of  their  squares  is  155.     Find  the  numbers. 

4.  There  are  three  numbers  in  arithmetical  progression  the 
sum  of  whose  squares  is  93.     If  the  third  is  4  times  as  large 

the  first,  what  are  the  numbers  ? 

5.  Find  the  sum  of  the  odd  numbers  1  to  99,  inclusive. 

6.  The  product  of  the  extremes  of  an  arithmetical  progres- 
oii  of  10  terms  is  70,  and  the  sum  of  the  series  is  95.     What 

are  the  extremes? 

7.  Fifty-five  logs  are  to  be  piled  so  that  the  top  layer  shall 
consist  of  1  log,  the  next  layer  of  2  logs,  the  next  layer  of  3 
logs,  etc.    How  many  logs  must  be  placed  in  the  bottom  layer  ? 

milnk's  stand,  alg.  —  26 


402  PROGRESSIONS 

8.  It  cost  Mr.  Smith  $  19.00  to  have  a  well  dug.  If  the 
cost  of  digging  was  f  1.50  for  the  first  yard,  $1.75  for  the 
second,  $  2.00  for  the  third,  etc.,  how  deep  was  the  well  ? 

9.  How  many  arithmetical  means  must  be  inserted  between 
4  and  25,  so  that  the  sum  of  the  series  may  be  IIG  ? 

10.  Prove  that  equal  multiples  of  the  terms  of  an  arith- 
metical progression  are  in  arithmetical  progression. 

11.  Prove  that  the  diiference  of  the  squares  of  consecutive 
integers  are  in  arithmetical  progression,  and  that  the  common 
difference  is  2. 

12.  Prove  that  the  sum  of  n  consecutive  odd  integers, 
beginning  with  1,  is  n'\ 

GEOMETRICAL    PROGRESSIONS 

519.  A  series  of  numbers  each  of  which  after  the  first  is 
derived  by  multiplying  the  preceding  number  by  some  con- 
stant multiplier  is  called  a  geometrical  series,  or  a  geometrical 
progression. 

2,  4,  8,  16,  32  and  a*,  a^,  a"^,  a  are  geometrical  progressions. 

In  the  first  series  tlie  constant  multiplier  is  2  ;  in  the  second  it  is  -  • 

G.P.  is  an  abbreviation  of  the  words  geometrical  progression. 

520.  The  constant  multiplier  is  called  the  ratio. 

It  is  evident  that  the  terms  of  a  geometrical  progression 
increase  or  decrease  numerically  according  as  the  ratio  is 
numerically  greater  or  less  than  1. 

521.  To  find  the  nth,  or  last,  term  of  a  geometrical  series. 

Let  a  represent  the  first  term  of  a  G.P.,  /•  the  ratio,  n  the 
number  of  terms,  and  I  the  last,  or  nth,  term. 

Then,  the  series  is  a,  ar,  ar^,  a)-^,  ar*,  •••. 

Observe  that  the  exponent  of  r  is  one  less  than  the  number 
of  the  term  ;  that  is, 

I  =  ar'-\  (I) 


PROGKESSIONS  403 

EXERCISES 

522.    1.    Find  the  yth  term  of  the  series  1,  3,  9,  •••. 

l*KOCKSS 

Explanation.  — In  this  exercise  a  =  1,  r  =  3,  and 

I  =  ar'-^  n  =  9. 

_  4       .,«  Substituting  tiiese  values  in  the  fonnula  for  I,  the 

~  ,.^     '  last  term  is  found  to  be  (J661. 

=  6501 

2.  Find  the  10th  term  of  the  series  1,  2,  4,  •••. 

3.  Find  the  8th  term  of  the  series  \f  ^,  1,  •••. 

4.  Find  the  9th  term  of  the  series  6, 12,  24,  •••. 

5.  Find  the  11th  term  of  the  series  i,  1,  2,  •••. 

6.  Find  the  7th  term  of  the  series  2,  6,  18,  •••. 

7.  Find  the  6th  term  of  the  series  4,  20,  100,  .••. 

8.  Find  the  6th  term  of  the  series  6,  18,  54,  •••. 

9.  Find  the  10th  term  of  the  series  1,  {,  J,  •••• 

10.  Find  the  10th  term  of  the  series  1,  J,  J,  ••♦. 

11.  Find  the  8th  term  of  the  series  i,  J,  |,  •••• 

12.  Find  the  11th  term  of  the  series  a^%  tn-,  •.-. 

13.  Find  the  wth  term  of  the  series  2,  V2,  1,  •••. 

14.  If  a  man  begins  business  with  a  capital  of  $2000  and 
•  loubles  it  every  year  for  6  years,  how  much  is  his  capital  at 
tlie  end  of  the  sixth  year? 

15.  The  population  of  the  United  States  was  76.3  millions 
ill  1900.  If  it  doubles  itself  every  25  years,  what  will  it  be  in 
tlie  year  2000? 

16.  A  man's  salary  was  raised  \  every  year  for  5  years.  If 
his  salary  was  $512  the  first  year,  what  was  it  the  sixth  year? 

17.  The  population  of  a  city,  which  at  a  certain  time  was 
L't»,736,  increased  in  geometrical  progression  25%  each  decade. 
What  was  the  population  at  the  end  of  40  years  ? 


404  progrp:ssions 

18.  A  man  who  wanted  10  bushels  of  wheat  thought  $  1  a 
bushel  too  high  a  price ;  but  he  agreed  to  pay  2  cents  for  the 
first  bushel,  4  cents  for  the  second,  8  cents  for  the  third,  and 
so  on.     How  much  did  the  last  bushel  cost  him  ? 

19.  The  machinery  in  a  manufacturing  establishment  is 
valued  at  ^20,000.  If  its  value  depreciates  each  year  to  the 
extent  of  10  %  of  its  value  at  the  beginning  of  that  year,  how 
much  will  the  machinery  be  worth  at  the  end  of  5  years  ? 

20.  From  a  grain  of  corn  there  grew  a  stalk  that  produced 
an  ear  of  150  grains.  These  grains  were  planted,  and  each 
produced  an  ear  of  150  grains.  This  process  was  repeated 
until  there  were  4  harvestings.  If  75  ears  of  corn  make  1 
bushel,  how  many  bushels  were  there  the  fourth  year? 


523.  A  series  consisting  of  a  limited  number  of  terms  is 
called  a  finite  series. 

524.  A  series  consisting  of  an  unlimited  number  of  terms  is 
called  an  infinite  series. 

525.  To  find  the  sum  of  a  finite  geometrical  series. 

Let  a  represent  the  first  term,  r  the  ratio,  n  the  number  of 
terms,  I  the  7ith,  or  last,  term,  and  s  the  sum  of  the  terms. 

Then,  s  =  a-^ar -\-ar^-\-ar^  +  •••  -j-ar"-\  (1) 

(1)  X  r,  rs  =  ar  +  ar^  +  ar^  +  •  •  •  +  ^r""^  +  ar\  (2) 

(2)-(l),     s(r-l)  =  ar»-a. 

(H) 


•.  s  = 

ar'' 

—  a 

r  - 

-1 

But,  since  ar 

n-l  _ 

I, 

ar''  = 

■  rl 

Substituting 

7'1  for 

ar'' 

in  (II), 

s  = 

rl  —  a 

or 

a- 
Y- 

-rl 
—r 

(III) 


n;()(;i{K8SioNS  405 

EXERCISES 

526.  1.    Find  the  sum  of  6  terms  of  the  series  3,  9,  27,  .-•. 

PKOCESS 

<(r"  —  a  ExpLAXATiON.  — Since  the  first  term  a,  the 

/  -_  1  ratio  r,  and  the  number  of  terms  «,  are  known, 

formula  II,  which  gives  the  sum  in  terms  of  a, 
_  o  X  o  —  o  _  1  AQo     r,  and  n,  is  used. 
3-1 

2.  Find  the  sum  of  8  terms  of  the  series  1,  2,  4,  •••. 

3.  Find  the  sum  of  8  terms  of  the  series  1,  |,  J,  •••. 

4.  Find  the  sum  of  10  terms  of  the  series  1,  1|,  2J,  •  •.. 

5.  Find  the  sum  of  7  terms  of  the  series  2,  —  },  f,  •••. 

6.  Find  the  sum  of  12  terms  of  the  series  —  i,  J,  —  i,  •••• 

7.  Find  the  sum  of  7  terms  of  the  series  1,  2  a?,  4  a^,  •••. 

8.  Find  the  sum  of  7  terms  of  the  series  1,  —  2  a?,  4  a:*,  ••♦. 

9.  Find  the  sum  of  n  terms  of  the  series  1,  ot^y  a;*,  •••. 

10.  Find  the  sum  of  n  terms  of  the  series  1,  2,  4,  •••. 

11.  Find  the  sum  of  n  terms  of  the  series  1,  ^,  J,    ••. 

12.  The  extremes  of  a  geometrical  series  are  1  and  729,  and 
t  he  ratio  is  3.     What  is  the  sum  of  the  series  ? 

13.  What  is  the  sum  of  the  series  3,  6,  12,    ••,  192? 

14.  What  is  the  sum  of  the  series  7,  •••,  -  56,  112,  -  224  ? 

527.  To  find  the  sum  of  an  infinite  geometrical  series. 

If  the  ratio  r  is  numerically  less  than  1,  it  is  evident  that 
tlie  successive  terms  of  a  geometrical  series  become  numeric- 
ally less  and  less.  Hence,  in  an  infinite  decreasing  geomet- 
rical series,  the  nth  term  I,  or  ar^~\  can  be  made  less  than  any 
assi£:jnable  number,  though  not  absolutely  equal  to  zero. 


406  PROGRESSIONS 

Formula  (III),  page  404,  may  be  written, 

_  _a t1__ 

^  ~l-r      1 - r 

Since  by  taking  enough  terms  I  and,  consequently,  rl  can  be 
made  less  than  any  assignable  number,  the  second  fraction  may 
be  neglected. 

Hence,  the  formula  for  the  sum  of  an  infinite  decreasing 
geometrical  series  is 

EXERCISES 

528.   1.   Find  the  sum  of  the  series  1,  J^,  y^,  •••. 

Solution 
Substituting  1  for  a  and  jL  for  r  in  (IV), 

Find  the  value  of : 

2.  H-i  +  i  +  ---. 

3.  3  +  I  +  A+-. 

8.  l  +  a;H-ar^4-a^  + 

9.  l  —  x-\-dL^  —  x^-\ ,  when  x='^. 

10.   Find  the  value  of  the  repeating  decimal  .185185185  •••. 

Solution 
Since    .185185185 •••  =  .185  +  .000185  +  .000000185  +  ■••,  a  =  .185  and 


5. 

-4- 

-1- 

i- 

6. 

-2+1- 

A  + 

7. 

100- 

- 10  -f  1  - 

when  X  = 

.9. 

= .001. 
Substituting  in  (IV),     .185185185 •• 

1 

.185          5 
-  .001      27 

Find  the  the  value  of : 

11.    .407407.... 

14. 

.020303.... 

12.    .363636... 

15. 

.007007... 

13.    1.94444.... 

16. 

5.032828.. 

PROGRESSIONS  407 

529.    To  insert  geometrical  means  between  two  terms. 

EXERCISES 

1.  Insert  3  geometrical  means  between  2  and  162. 

PROCESS  Explanation.  —  Since  there  are  three  means,  there  are 

;_j^^-i  five  terms,  and  «  —  1  =  4.     Solving  for  r  and  neglecting 

1 1">  _  9   4  imaginary  values,  r=±3. 

ih^  —  ^r  Therefore,  the  series  is  either  2,  6,  18,  64,  162  or  2,  -  0, 

r=±3  18,-64,162. 

2.  Insert  3  geometrical  means  between  1  and  625. 

3.  Insert  5  geometrical  means  between  4 J  and  ^f  f  ^. 

4.  Insert  4  geometrical  means  between  ^^  and  |J. 

5.  Insert  4  geometrical  means  between  5120  and  5. 

6.  Insert  4  geometrical  means  between  4\/2and  1. 

7.  Insert  5  geometrical  means  between  a'  and  b^. 

8.  Insert  4  geometrical  means  l)etween  x  and  —  ?/. 


530.  If  (r  is  the  geometrical  mean  between  a  and  6,  in  the 
'""''"  a,  O,  b, 

by  §519,  ^  =  ^. 

G=  ±  ^ab.     That  is, 

Principle.  —  Tfie  geometrical  mean  between  two  numbers  is 
c(jual  to  the  square  root  of  their  product. 

Observe  that  the  geometrical  mean  between  two  numbers  is  also  their 
mean  proportional. 

EXERCISES 

531.  Find  the  geometrical  mean  between: 

1.  8  and  50.  4.    (a  +  bf  and  (a  -  by. 

2.  .V  and  SL  2  ,     u  u  ,  u2 

K    ^  H-qfe  on,i  abz\-b^ 

3.  Ijiandf.  ^-   a'^aJb         ab-b^' 


408  PROGRESSIONS 

532.  Since  formula  I  with  formula  II,  or  III,  which  is 
equivalent  to  II,  forms  a  system  of  two  independent  simulta- 
neous equations  containing  five  elements,  if  three  elements  are 
known,  the  other  two  may  be  found  by  elimination. 

Note.  —  Solving  for  w,  since  it  is  an  exponent,  requires  a  knowledge 
of  logarithms  (§§  558-598),  except  in  cases  where  its  value  may  be  deter- 
mined by  inspection.     Only  such  cases  are  given  in  this  chapter. 

Problems 

533.  1.    Given  r,  I,  and  s,  to  find  a. 

2.  The  ratio  of  a  geometrical  progression  is  5,  the  last  term 
is  625,  and  the  sum  is  775.     What  is  the  first  term  ? 

3.  The  ratio  of  a  geometrical  progression  is  y\,  the  sum  is 
\,  and  the  series  is  infinite.     What  is  the  first  term  ? 

4.  Find  I  in  terms  of  a,  r,  and  s. 

5.  Find  the  last  term  of  the  series  5,  10,  20,  •••,  the  sum  of 
whose  terms  is  155. 

6.  If  1  4-  iV2  +  i  +  ••■  =  If  (1  +  V2),  what  is  the  last 
term,  and  the  number  of  terms  ? 

7.  Deduce  the  formula  for  r  in  terms  of  a,  I,  and  s. 

8.  If  the  sum  of  the  geometrical  progression  32,  •••,  243  is 
Q%^,  what  is  the  ratio  ?     Write  the  series. 

9.  The  sum  of  a  geometrical  progression  is  700  greater  than 
the  first  term  and  525  greater  than  the  last  term.  What  is  the 
ratio  ?     If  the  first  term  is  81,  what  is  the  progression  ? 

10.  Deduce  the  formula  for  r  in  terms  of  a,  n,  and  L 

11.  The  first  term  of  a  geometrical  progression  is  3,  the  last 
term  is  729,  and  the  number  of  terms  is  6.  What  is  the  ratio? 
Write  the  series. 

12.  Find  I  in  terms  of  r,  n,  and  s. 

13.  The  velocity  of  a  sled  at  the  bottom  of  a  hill  is  100  feet 
per  second.  How  far  will  it  go  on  the  level,  if  its  velocity 
decreases  each  second  ^  of  that  of  the  previous  second  ? 


PROGRESSIONS  409 

14.  Under  normal  conditions  the  members  of  a  certain 
species  of  bacteria  reproduce  by  division  (each  individual  into 
two)  every  half  hour.  If  no  hindrance  is  offered,  how  many 
biicteria  will  a  single  individual  produce  in  8  hours  ? 

15.  A  ball  thrown  vertically  into  the  air  100  feet  falls  and 
rebounds  40  feet  the  first  time,  16  feet  the  second  time,  and  so 
on.  What  is  the  whole  distance  through  which  the  ball  will 
have  passed  when  it  finally  comes  to  rest  ? 

16.  Show  that  the  amount  of  $1  for  1,  2,  3,  4,  5  years  at 
compound  interest  varies  in  geometrical  progression. 

17.  Show  that  equal  multiples  of  numbers  in  geometrical 
progression  are  also  in  geometrical  progression. 

18.  The  sura  of  three  numbers  in  geometrical  progression  is 
19,  and  the  sum  of  their  squares  is  133.     Find  the  numbers. 

SuoGKSTiox.  —  When  there  are  but  three  terms  in  the  series,  they  may 
be  represented  by  a;2,  xy,  y*-*,  or  by  x,  Vicy,  y. 

19.  The  product  of  three  numbers  in  geometrical  progres- 
sion is  8,  and  the  sum  of  their  squares  is  21.  What  are  the 
three  numbers  ? 

20.  The  sum  of  the  first  and  second  of  four  numbers  in  geo- 
metrical progression  is  15,  and  the  sum  of  the  third  and  fourth 
is  GO.     What  are  the  numbers  ? 

Suggestion.  —  Four  unknown  numbers  in  geometrical  progression  may 

li»'  represented  by  — ,  ar,  y,  ^. 
y  X 

21.  From  a  cask  of  vinegar  \  was  drawn  off  and  the  cask 
w;is  filled  by  pouring  in  water.  Show  that  if  this  is  done  (> 
times,  the  contents  of  the  cask  will  be  more  than  -fj^  water. 

22.  If  the  quantity,  and  correspondingly  the  pressure,  of 
the  air  in  the  receiver  of  an  air  pump  is  diminished  by  -^  of 
itself  at  each  stroke  of  the  piston,  and  if  the  initial  pressure  is 
14.7  pounds  per  square  inch,  find,  to  the  nearest  tenth  of  a 
pound,  what  the  pressure  will  be  after  6  strokes. 


410  PROGRESSIONS 

23.  A  man  bought  a  farm  for  $5000,  agreeing  to  pay  prin- 
cipal and  interest  in  five  equal  annual  installments.  Find  the 
annual  payment,  interest  at  6  %. 

SOLUTIOX 

By  the  conditions  of  the  problem  the  equal  payments  are  to  include  the 
interest  accrued  at  the  end  of  each  year  plus  a  portion  of  the  principal. 

The  principal  for  the  second  year  will  be  less  than  the  principal  for  the 
first  year  by  the  portion  of  the  principal  paid  at  the  end  of  the  first  year ; 
therefore,  the  interest  to  be  paid  at  the  second  payment  will  be  less  than 
the  interest  paid  at  the  first  payment  by  the  interest  for  1  year  upon  the 
first  portion  of  the  principal  paid,  or  6%  of  the  portion  of  the  principal 
paid  the  first  year. 

Since  the  payments  are  to  be  equal,  the  portion  of  the  principal  to  be 
paid  at  the  second  payment  must  be  as  much  more  than  the  portion  paid 
at  the  first  payment  as  the  interest  is  less  than  the  interest  paid  at  the  first 
payment ;  that  is,  it  must  be  6%  more  than,  or  1.06  of,  the  portion  of  the 
principal  first  paid. 

By  reasoning  in  the  same  way  regarding  subsequent  payments,  the 
third  portion  of  the  principal  paid  will  be  found  to  be  1.06  of  the  second, 
the  fourth  1.06  of  the  third,  and  the  fifth  1.06  of  the  fourth  ;  that  is,  the 
portions  of  the  principal  paid  form  a  G.P.,  in  which  r  =  1.06,  n  =  b  (the 
number  of  payments),  and  8  =  $5000.     We  desire  to  find  a. 

Substituting  the  known  values  in  (I)  and  (III),  we  have 

Z  =  al.065-i,  or  Z  =  1.064  a;  (1) 

and  .     $5000  =  ^'Q^^-^orZ^^  +  •^^^Q^X'Q^  (2) 

1.06-1  1.06  ^  ^ 

Eliminating  I  from  (1)  and  (2)  and  solving  for  «,  we  have 

$5000x.06^       gg^gg^^ 
1.065  _  1 

That  is,  the  first  portion  of  the  principal  paid  =  -8886.98  ;  but  the  first 
year's  interest  =6%  of  $5000,  or  -$300  ;  hence,  the  entire  first  payment 
=  $886.98  +  $300  =  $  1186.98,  which  is  also  each  annual  payment. 

24.  A  man  borrowed  $  loOO,  agreeing  to  pay  principal  and 
interest  at  6  %  in  four  equal  annual  installments.  Find  the 
sum  to  be  paid  each  year. 

25.  A  father  bequeathed  to  his  son  $  10,000,  the  bequest  and 
interest  at  4  %  to  be  paid  in  six  equal  annual  installments. 
Find  the  annual  payment. 


INTERPRETATION  OF  RESULTS 


534.  A  number  that  has  the  same  value  throughout  a  dis- 
cussion is  called  a  constant. 

Arithmetical  numbers  are  constants.  A  literal  number  is  constant  in 
a  discussion,  if  it  keeps  the  same  value  throughout  that  discussion. 

535.  A  number  that  under  the  conditions  imposed  upon  it 
may  have  a  series  of  different  values  is  called  a  variable. 

The  numbers  .3,  .33,  .333,  .3333,  ...  are  successive  values  of  a 
variable  approaching  in  value  the  constant  \. 

536.  When  a  variable  takes  a  series  of  values  that  approach 
nearer  and  nearer  a  given  constant  without  becoming  equal  to 
it,  so  that  by  taking  a  sufficient  number  of  steps  the  difference 
between  the  variable  and  the  constant  can  be  made  numerically 
less  than  any  conceivable  numl^er  however  small,  the  constant 
is  called  the  limit  of  the  variable,  and  the  variable  is  said  to 
approach  its  limit. 

This     figure     represents 

iiphically  a  variable  x  ap-      o v y,      ?i  .  .^ 

i'loaching  its  limit  0X=2.  1  4  i     i 

The    first     value     is    0X\ 

=  1  ;  the  second  is  OX2  =  IJ ;  the  third  is  OX3  =  1}  ;  etc. 

At  each  step  the  difference  between  the  variable  and  its  limit  is 
•  liminished  by  half  of  itself.  Consequently,  by  taking  a  sufficient  number 
<  >f  steps  this  difference  may  become  less  than  any  number,  however  small, 
tliat  may  be  assigned. 

537.  A  variable  that  may  become  numerically  greater  than 
.uiy  assignable  number  is  said  to  be  infinite. 

The  symbol  of  an  infinite  number  is  x  . 

411 


412  INTERPRETATION   OF   RESULTS 

538.  A  variable  that  may  become  numerically  less  than  any 
assignable  number  is  said  to  be  infinitesimal. 

An  infinitesimal  is  a  variable  whose  limit  is  zero. 

The  character  0  is  used  as  a  symbol  for  an  infinitesimal  num- 
ber as  well  as  for  absolute  zero,  which  is  the  result  obtained  by 
subtracting  a  number  from  itself. 

539.  A  number  that  cannot  become  either  infinite  or  infini- 
tesimal is  said  to  be  finite. 

THE   FORMS   a  X  0,  ?,    ^,    ^  ,    ?     ^ 
a      0     00      0     00 

540.  The  results  of  algebraic  processes  may  appear  in  the 

forms,  a  X  0,  -,   -,_,-,  ^  ,  etc.,  which  are  arithmetically 
a    0    00     0    GO 

meaningless ;  consequently,  it  becomes  important  to  interpret 

the  meaning  of  such  forms. 

541.  Interpretation  of  ff  x  0. 

1.  Let  0  represent  absolute  zero,  defined  by  the  identity, 

0  =  n-n.  (1) 

Multiplying  a  =  a  by  (1),  member  by  member,  Ax.  3,  we  have 
a  X  0  =  a(n  —  n) 
=  an  —  an 
by  def.  of  zero,  =  0.     That  is. 

Any  finite  number  multiplied  by  zero  is  equal  to  zero. 

2.  Let  0  represent  an  infinitesimal,  as  the  variable  whose 
successive  values  are  1,  .1,  .01,  .001,  •••. 

Then,  the  successive  values  of  a  x  0  are  (§  81) 

a,  .1  a,  .01  a,  .001  a,  •••.      Hence, 
a  X  0  is  a  variable  whose  liinit  is  absolute  zero.     That  is. 

Any  finite  number  multiplied  by  an  infinitesimal  number  is 
equal  to  an  infinitesimal  number. 


liN TERPRETATION  OF   RESULTS  413 


542.   Interpretation  of  -. 
a 

1.    Let  0  represent 

absolute  zero,  defined  by  the  identity, 

0  =  n  -  n. 

Dividing  by  a, 

0_»      n. 
a     a     a^ 

but  by  def.  of  zero, 

a     a 

Hence,  Ax.  5, 

-  =  0.    That  is, 
a 

Zero  divided  by  any  finite  nuvnber  is  equal  to  zero. 

2.   Let  0  represent  an  ivfinitesimalj  as  the  variable  whose 
successive  values  are  1,  .1,  .01,  .001,  •••. 

Then,  the  successive  values  of  -  are    -,   — ,   ^ — ,   * ,  •••; 

a  a     a      a        a 

wlience,  -  is  a  variable  whose  limit  is  absolute  zero. 
a 

A  HI/  injiuitesimal  number  divided  by  a  finite  number  is  equal  to 
II II  infinitesimal  number. 

543.    Interpretation  of  ^. 

The  successive  values  of  the  fractions,  -,    — ,   — ,    ,  etc., 

2     .2     .02    .002 

are  .5,  5,  50,  500,  etc.,  and  they  continually  increase   as  the 
nominators  decrease. 
In  general,  if  the  numerator  of  the  fraction  -  is  constant  while 

X 

tlie  denominator  decreases  regularly  until  it  becomes  nuraer- 
i<  ally  less  than  any  assignable  number,  the  quotient  will  in- 
crease regularly  and  become  numerically  greater  than  any 
assignable  number. 

.•.-=«.     That  is, 
0 

If  a  finite  number  is  divided  by  an  infinitesim^al  number,  tJie 

quotient  will  be  an  infinite  number. 


414  INTERPKETATION  OF   RESULTS 


544.  Interpretation  of  — . 

QO 

The  successive  values  of  the  fractions,  -,    — ,    -— ,  , 

'  2'  20'  200'  2000' 
etc.,  are  .5,  .05,  .005,  .0005,  etc.,  and  they  continually  decrease 
as  the  denominators  increase. 

In  general,  if  the  numerator  of  the  fraction  -  is  constant 

X 

while  the  denominator  increases  regularly  until  it  becomes 
numerically  greater  than  any  assignable  number,  the  quotient 
vi^ill  decrease  regularly  and  become  numerically  less  than  any 
assignable  number. 

.-.   -=0.     That  is, 

00 

If  a  finite  number  is  divided  by  an  infinite  number,  the  quotient 
will  be  an  infinitesimal  number. 

545.  Interpretation  of  -. 

Let  0  represent  absolute  zero. 

Then,  if  a  is  any  finite  number,  §  541, 

a  X  0  =  0; 

whence,  -  =  a.     That  is, 

0 

When  0  represents  absolute  zero,  -  is  the  symbol  of  an  indeter- 
minate number. 

546.  Interpretation  of  —  . 

oo 

Let  a  represent  any  finite  number  and  x  any  number  what- 
ever. 

a 

Then,  ±  =  ^.^  =  a.  (1) 

\       X     1 

X 

If  X  decreases  regularly  until  it  becomes   numerically  less 
than  any  assignable  number  (§  543),  -  and  -  each  become  oc  . 


INTKRlTiETATION    <)I     KKSl'LTS  415 


^     Consequentls ,  i^l ;  becomes  —  =  a,  any  finite  nuniher. 
Ilence,  —is  the  symbol  of  an  indeterminate  mimber. 

547.  Since  (§  543)  -  is  infinite  and  (§  545)  -  is  indetermi- 
nate, it  is  seen  that  axiom  4  (§  68)  is  not  applicable  when 
the  divisor  is  0;  that  is,  it  is  not  allowable  to  divide  by 
absolute  zero. 

The  student  may  point  out  the  inadmissible   step  or   fal- 

^^^^'>'"^=  7  ..-35  =  3  0.-15, 

7(x  -  5)  =  S(x  -  5). 
.-.  7  =  3. 
Sr<;«;KSTioN.  —  Solve  the  equation  to  find  what  divisor  has  been  used. 

548.  Fractions  indeterminate  in  form. 

Some  fractions,  for  certain  values  of  the  variable  involved, 

give  the  result  9,  which,  however,  is  indeterminate  only  infoi-my 

because  a  definite  value  for  the  fraction  may  often  be  found. 

For  example,  when  x  =  1.  by  substituting  directly,  ^"  ~     =  -• 

3C  —  1       0 

rhough  ^  ~  ^  =  -(?-±-lK£ziil  =  a:  +  1 ,  it  is  not  allowable  to  perform 
X  -  1  X  —  1 

this  operation  in  finding  the  value  of  the  fraction  when  x  =  1,  that  is, 

when  X  —  1  =  0,  for  (§  647)  it  is  not  allowable  to  divide  by  absolute  zero. 

II  .wever,  since  the  value  of  ^  ~     is  always  the  same  as  the  value  of 

X-  I 

1  so  long  asx  :?!:  1,  let  X  approach  1  as  a  limit. 
liut  (§  530)  X  cannot  become  1,  and  it  is  allowable  to  divide  by  x  -  1. 

~2_  1 

Now  as  X  approaches  1  as  a  limit, approaches  x  +  1,  or  2,  as  a 

X  —  I 

limit,  and  so  2  is  called  the  value  of  the  fraction.    That  is, 

The  value  of  such  a  fraction  for  any  given  value  of  the  vari- 
able involved  is  the  limit  that  the  fraction  approaches  as  the 
variable  approaches  the  given  value  as  its  limit. 


THE   BINOMIAL   THEOREM 


549.  The  Binomial  Theorem  derives  a  formula  by  means  of 
which  any  indicated  power  of  a  binomial  may  be  expanded  into 
a  series. 

POSITIVE    INTEGRAL   EXPONENTS 

550.  By  actual  multiplication, 

(a  +  xY  =  a^  -\-2ax  -\-  x^. 

(a  -j-xf  =  a^  +  ^  a'x  +  3  ax^  +  x^ 

(a -\- xy  =  a^ -\- 4  a^x  +  6  a'^x-  -j-  4  ax'  +  x'^. 

These  powers  of  (a  +  x)  may  be  written,  respectively : 


(a  +  xf  =  a-  +  2  ax  + 
(a  -\-  xy  =  a^  +  3  a^x  -f 


{a  +  xy  =  a^  4-  4  a!^x  +  - 


—  ax^  H XT. 

2  1.2.3 

3,2.2^4. 3. 2^^  ,  4.3.2.1    , 


2  1-2.3 


aa?  -\ x^. 

1.2.3.4 


If  the  law  of  development  revealed  in  the  above  is  assumed 
to  apply  to  the  expansion  of  any  power  of  any  binomial,  as  the 
nth  power  of  {a  -f-  x),  the  result  is 

(a  +  xy 


1.2.3 


.an-3^3_j_...     (I) 


From  formula  (I)  it  is  evident  that  in  any  term : 
1.    The  exponent  of  a;  is  1  less  than  the  number  of  the  term. 
Hence,  the  exponent  of  x  in  the  (>•  +  l)th  term  is  r. 

416 


THE  BINOMIAL  THEOREM  417 

2.   The  exponent  of  a  is  n  minus  the  exponent  of  x. 

Hence,  the  exponent  of  a  in  the  (r  +  l)th  term  is  n  —  r. 

.').  The  number  of  factors  in  the  numerator  and  in  the 
denominator  of  any  coetticient  is  1  less  than  the  number  of  the 
term. 

Hence,  the  coefficient  of  the  (r  +  l)th  term  has  r  factors  in 
the  numerator  and  r  factors  in  the  denominator. 

Therefore,  the  (r  -f  l)th,  or  general,  term,  is 

n(?i-l)(n-2)  ...  to  r  factors^,,, ^,  ^. 

1  .2.3. ••  tor  factors  *  ^ 

When  there  are  two  factors  in  the  numerator,  the  last  is 
n  —  1 ;  when  there  are  three  factors,  ?i  —  2;  when  there  are  four 
fa(*tors,  n  —  3,  etc.  Therefore,  when  there  are  r  factors,  the 
last  is  ?i  —  (r  —  1),  or  w  —  r  H-  1.     Hence,  (1)  may  be  written 

»(n -!)(«- 2) -(»-r  +  l)„._,^  (2) 

Pherefore,  the  full  form  of  (I)  is 

"  tl'  +  „a-x  +  !^!^L)„.-V  +  «(n-l)(»-2)  ^^^  _^ 
L  '  2  1  •  2  •  3 

^»(n-l)(n-2) -(»-r  +  l)  ^^^^     .  ^^ 

This  is  called  the  binomial  formula.  It  will  now  be  proved 
to  be  true  for  positive  integral  exponents. 

551.  Since  it  has  already  been  proved,  by  actual  multiplica- 
1  (§  550),  that  the  binomial  formula  is  true  for  the  second^ 
tinrdj^xidi  fourth  powers  of  a  binomial,  it  remains  to  discover 
whether  it  is  true  for  powers  higher  than  the  fourth. 

If  the  binomial  theorem,  when  assumed  to  be  true  for  the 
uth  power,  can  be  proved  to  be  true  for  the  (n  -f  l)th  power, 
since  it  is  known  to  be  true  when  the  nth  power  is  the  fourth 
power,  it  will  then  have  been  proved  to  be  true  for  the  fifth 
power;  also  for  the  sixth  power,  being  true  for  the  fifth  power ; 
and  in  like  manner  for  each  succeeding  power. 

MII.NK'8    STAND.     AI.G.  27 


418 


THE   BINOMIAL   THEOREM 


Therefore,  it  remains  to  prove  that  if  (I)  is  true  for  the  nth. 
power,  it  will  hold  true  for  the  (n  +  l)th  power. 

The  (n  +  l)th  power  of  (a  -\-  x)  may  be  obtained  from  the 
nth  power  by  multiplying  both  members  of    (I)  by  {a  -\-  x). 

Then,  we  have 
(a_|_;^)«  +  i 


la'^  +  ^  +  n 


+1 


„     ,  n(n  —  l) 
1.2 


^„_, ^.2  ,n(n-l)(n-2) 
'  1.2-3 

n(7i—l) 


a"-^x--j- 


1-2 


Collecting  the  coefficients  of  like  powers  of  a  and  x,  we  have 
Coefficient  of  a"  a;      =  n  +1. 

Coefficient  of  a""  V  =  !?&:il)  4.  n 

_7i^  —  7i-\-2n_  (n  +  1  )n 


1  .2 


1  .2 


^  Coefficient  of  a^^.r^  ^  K!Lzl1X!?^1^.  _^  K"-^) 

1.2.3  1.2 


1.2.  3 


(?j. +  1)  ?i  (n-1) 
1.2.3 


...  (a  +  a;)"  +  ^=a"  +  i  +  (y/4-lK.^  +  ^^^-=tl>a"-V 


1  .2 


4-^^^+^|f|^a"-.r^4-.--. 


(11) 


Upon  comparison  it  may  be  seen  that  (II)  and  (I)  have  the 
same  form,  n  +  1  in  one  taking  the  place  of  n  in  the  other. 
That  is,  (II)  and  (I)  express  the  same  law  of  formation. 

Therefore,  if  the  formula  is  true  for  the  7ith  power,  it  holds 
true  for  the  (n  -f  l)th  power. 

By  actual  multiplication  (§  550)  the  formula  is  known 
to  be  true  for  the  fourth  power.     Consequently,  it  is  true  for 


Tin:    BINOMIAL   THEOREM  419 

the Jijlh  power;  aiid  then  being  true  for  the  Jijlh  power,  it  is 
true  for  the  sixth  power;  aud  so  on  for  each  succeeding  power. 

Hence,  the  binomial  formula  is  true  for  any  positive  integral 
exponent. 

This  proof  is  known  as  a  proof  by  mathematical  induction. 

552.  If  —  ic  is  substituted  for  x  in  (I),  the  terms  that  con- 
tain tiie  odd  powers  of  —  a;  will  be  negative,  and  those  that 
contain  the  even  powers  will  be  positive.     Therefore, 

^'*~^^"    .-I      ,n(n-l)    ,_....      n(n-l)(n-2)   ,_3  ,        ,tttn 

-   rr  —  na'^^x  H — ^^ -a"  -XT ^> 1^ -a""  "^x^  •••.  (Ill) 

1-2  1.2-3  ^ 

If  a  =  1,  (I)  becomes 

(i+.r  =  i  +  "^+'^^f^^+''^''~y3~^>^+----    (IV) 

553.  From  (I)  it  is  seen  that  the  last  factor  in  the  numera- 
tor of  the  coefficient  is  n  for  the  2d  terra,  n  —  1  for  the  3d 
term,  ?t  — 2  for  the  4th  term,  n  — (/i  — 2),  or  2,  for  the  7ith 
term,  and  7t —  («  — 1),  or  1,  for  the  (?i-hl)th  term;  and  that 
the  coefficient  of  the  (n  +  2)th  term,  and  of  each  succeeding 
term,  contains  the  factor  n  —  «,  or  0,  and  therefore  reduces  to  0. 
Hence, 

When  n  is  a  positive  integer,  the  series  formed  by  exjtandtJig 
I  -h  x)*  is  finite  and  has  n  -\-\  terms. 

554.  By  formula  (I)  when  n  is  a  positive  integer, 

(<(  -h  j:)- 

n   I        ..-1      I   n(n  —  l)    „_.,  9  ,  ,  «(n  — 1)'-«2  •  1  ^ 

11:  \    1--  (?i  — l))i 

'*^'*'"^^«>     ^\     .«(«-l'^-jt.        ,  «U-l)---2-l   , 
1-2  1.2  ••.(»  — l)/t 

A  comparison  of  the  two  series  shows  that : 
The  coefficients  of  the  latter  half  of  the  ex/mnsion  of  (a  -f-x)", 
'>''n  n  is  a  positive  integer,  are  the  sajne  as  those  of  the  first 
'/,  icritten  in  the  reverse  order. 


420  THE  BINOMIAL   THEOREM 

EXERCISES 

555.  1.   Expand  (Sa-2  by. 

Solution,  —  Substituting  S  a  tor  a,  2  b  for  x,  and  4  for  n  in  (IH), 
(3  a  -  2  6)4  =  (3  ay  -  4  (3  «)3(2  6)  +  ^  (3  ay (2  by 

=  81  a*  -  216  a^b  +  216  ^252  _  95  ^b^  +  iq  54. 
2.    Expand  f^  +  6a;Y. 
Suggestion.  — Since  (^-^bxY=[^(\  +  2  x)Y= -^(1  +  2  x)5, 

1)5 

(1  +  2  rK)5  maybe  expanded  by  (IV),  and  the  result  multiplied  by  —  ■ 
Expand : 

3.  (b-7iy. 

5.  (2-Sxy. 

6.  (x^-xy. 

7.  (a;H-ir-i)«. 

8.  (2a-\-Vxy. 

9.  (a  +  aVa)^ 

556.  To  find  any  term  of  the  expansion  of  (a  +  xy. 

Any  term  of  the  expansion  of  a  power  of  a  binomial  may 
be  obtained  by  substitution  in  (1)  or  (2),  §  550. 

In  the  expansion  of  a  power  of  the  difference  of  two  numbers  (a  -  x)", 
since  the  exponent  of  x  in  the  (r  +  l)th  term  is  r,  the  sign  of  the  general 
term  is  +  if  r  is  even,  and  —  if  r  is  odd. 


10. 

(-!)• 

15. 

(^^-^5)' 

11. 

(MJ- 

16. 

n-1              1 

{x  '^  -  xy. 

17. 

(ax-^-bVxy. 

12. 

e-9" 

18. 

13. 

{^a'+-\/¥y. 

14. 

(2-^2 -V^y. 

19. 

evi-ivi 

THE  BINOMIAL   THEOKEM  421 

EXERCISES 

557.    1.    Find  the  12th  term  of  (a  -  6)", 

Solution  • 

12th  term  =  14.13.12.11.10.9.8.7.6.6.4  ^3  ,, 

1.2.3.4.6.6.7.8.9.10.11         "^        ^ 

=  _  ^4  •  13 .  12  ^^1,  =  _  304  amK 
1.2.3 
Or,  since  there  are  16  terms,  the  coefficient  of  the  12th  term,  or  the  4th 
term  from  the  end,  is  equal  to  that  of  the  4th  term  from  the  beginning. 

.-.  12th  term  =  -  ^^'^^'^^  a«6"  =  -  364  a'fe" 
1.2.3 

Without  actually  expanding,  find  the : 

2.  4th  term  of  (a  +  2)^  5.    20th  term  of  (1  +  x)^. 

3.  8th  term  of  (x  -  .y)".  6.    18th  term  of  (1  -  2  aj)» 

4.  5th  term  of  (a;  -  2  y)".        7.    13th  term  of  (a*  -  a-^- 

8.  Find  the  middle  term  of  (a  +  3  6)*. 

9.  Find  the  6th  term 


10.  Find  the  middle  term 

11.  Find  the  two  middle  terms 


12.  In  the  expansion  of  (aj*  -f  x)",  find  the  term  containing  a^. 
SoLL'TioN.  —  Since   (gfl  +  a;)"  =  [x:^  ^1  +  -^  1"  =  a:«  ^1  -»-  - y\  every 

!•  nn  of  the  series  expanded  from  ( 1  +  -  )     will  be  multiplied  by  x^. 

^       ""^  /1\7         V 

Hence,  the  term  sought  is  that  which  contains  |  -  V  ,  or  — ;  that  is,  the 

(7  +  l)th,  or  8th  term.  ^*'  * 

8th  term  =  ^«11  •  10'«-8 /ly  ^  33^^,, 

1.2.3.4  w 

13.  Find  the  coefficient  of  a*  in  the  expansion  of  (a*  +  a)*. 

14.  Find  the  term  containing  a"6*  in  the  expansion  of  (a  —  6)" 
and  obtain  a  simple  expression  for  it  when  6  =  15  •(143*  a*)  '. 


422  THE  BINOMIAL  THEORKM 

The  binomial  formula  is  true  for  the  expansion  of  a  binomial 
when  the  exponent  is  negative  or  fractional,  provided  the  first 
term  of  the  binomial  is  numerically  greater  than  the  second. 
In  such  cases  the  expansion  is  an  infinite  series.  For  a  more 
extended  treatment  of  this  subject,  see  the  author's  Advanced 
Algebra. 

15.  Expand  (1  —  y)~'^  and  find  its  {r  +  l)th  term. 
Solution.  —  Substituting  1  for  a,  y  for  x,  and  —  1  for  n  in  (III), 

=  l+2/  +  2/2  +  ^3+  .... 
The  (r  +  l)th  term  is  evidently  y. 

Since  (1  — ?/)-i= ,  the  above  expansion  of  {I  —  y)-^  may  be 

verified  by  division.      ~  ^ 

16.  Expand  (a  +  x)^  to  five  terms  and  find  the  10th  term. 

Expand  to  four  terms  : 

17.  {l-a)-\  21.    {a^h)^.  25.    {l-\-xf. 

18.  (l  +  a)-\  22.    </{a-hY.  26.    {l-xy\ 


19.  (a  — by.  23.    V(9-a;)^.  27.    (a^  —  x~^y- 

20.  V4  +  X.  24.    {a-\-b)~K  28.    (a^  —  x^yt 

29.  Find  the  square  root  of  24  to  three  decimal  places. 
Solution.     v24  =  (24)^  =  (25  -  1)^  =  (25)^(1  -  ^^^  =  5(1-  ^i^)' 

L        2V25y         1-2     \2oJ  1.2.3        \2oJ  J 

^  5  _  .1  _  .001  -  .00002 =  4.89898  -  =  4.899,  nearly. 

Find,  to  three  decimal  places,  the  value  of : 

30.  V5.  32.    V26.  34.    -^9. 

31.  Vrr.  33.    ^/25.  35.    \/3(). 


LOGARITHMS 


558.  Early  in  the  seventeenth  century  a  scheme  was  devised 
to  simplify  long  computations  by  representing  all  real  positive 
numbers  as  powers  of  some  particular  number.  The  exponents 
of  these  powers,  called  logarithms^  were  arranged  in  tables  for 
convenient  reference ;  and  in  accordance  with  the  priuciples 
of  exponents,  multiplication  was  replaced  by  addition,  division 
by  subtraction,  involution  by  a  single  simple  multiplication, 
and  evolution  by  a  single  simple  division. 

Lord  Napier,  a  Scotchman,  was  the  inventor  of  logarithms  and 
hp  published  the  first  tables,  but  to  Henry  Briggs  belongs  the 
honor,  next  to  Napier,  for  their  development.  He  and  Napier 
independently  thought  of  the  advantage  of  a  system  that  would 
represent  all  numbers  as  powers  of  10  to  be  used  with  our 
decimal  system  of  notation,  but  after  consultation  with  each 
other  and  because  of  Napier's  declining  health,  it  was  left  to 
Briggs  to  work  out  the  system  that  is  in  common  use. 

559.  The  exponent  of  the  power  to  which  a  fixed  number, 
called  the  base,  must  he  raised  in  order  to  produce  a  given  num- 
l)t'r  is  called  the  logarithm  of  the  given  number. 

When  2  is  the  base,  the  logarithm  of  8  is  3,  for  8  =  2». 

When  10  Is  the  base,  the  logarithm  of  100  is  2,  for  100  =  10^  ;  the  loga- 
rithm of  1000  is  3,  for  1000  =  10" ;  the  logarithm  of  10,000  is  4,  for 
10,000  =  10*. 

560.  When  a  is  the  base,  x  the  exponent,  and  m  the  given 
number,  that  is,  when  a'  =  wi,  x  is  the  logarithm  of  the  number 
m  to  the  base  a,  written  log^  m  =  x. 

When  the  base  is  10,  it  is  not  indicated.  Thus,  the  logarithm  of  100  to 
xho  base  10  is  2.     It  is  written,  log  100  =  2. 

423 


424  LOGARITHMS 

561.  Logarithms  may  be  computed  with  any  arithmetical 
number  except  1  as  a  base,  but  the  base  of  tlie  common,  or 
Briggs,  system  of  logarithms  is  10. 

Since  10"   =  1,  the  logarithm  of  1  is  0. 
Since  10^   =  10,  the  logarithm  of  10  is  1. 
Since  10-   =  100,  the  logarithm  of  100  is  2. 
Since  10«   =  1000,  the  logarithm  of  1000  is  3. 
Since  10~^  =  J^,  the  logarithm  of  .1  is  —  1. 
Since  10~-  =  yi^,  the  logarithm  of  .01  is  —  2. 

562.  It  is  evident,  then,  that  the  logarithm  of  any  number 
between  1  and  10  is  a  number  greater  than  0  and  less  than  1. 
For  example,  the  logarithm  of  4  is  approximately  0.6021. 

Again,  the  logarithm  of  any  number  between  10  and  100  is 
a  number  greater  than  1  and  less  than  2.  For  example,  the 
logarithm  of  50  is  approximately  1.6990. 

Most  logarithms  are  endless  decimals.  All  the  laws  estab- 
lished for  other  exponents  apply  also  to  logarithms,  but  the 
proofs  have  been  omitted  as  being  too  difficult  for  the  beginner. 

563.  The  integral  part  of  a  logarithm  is  called  the  character- 
istic ;  the  fractional  or  decimal  part,  the  mantissa. 

In  log  50  =  1.6990,  the  characteristic  is  1  and  the  mantissa  is  .6990. 

564.  The  following  illustrate  characteristics^  mantissas,  and 

their  significance : 

log  4580     =  3.6609 ;  that  is,  4580     =  lO^^^^. 
log  458.0   =  2.6609 ;  that  is,  458.0    =  10=^«*^. 
log  45.80    =1.6609;  that  is,  45.80    =  10i«609. 
log  4.580   =  0.6609 ;  that  is,  4.580    =  W"^. 
log  .4580   =1.6609;  that  is,  .4580   =10-^  +  -^. 
log  .0458    =2.6609;  that  is,  .0458    =10-2+«^. 
log  .00458  =  3.6609 ;  that  is,  .00458  =  10 -»+  ««». 
From  the  above  examples  it  is  evident  that ; 


LOGARITHMS  425 

565.  Principles. —  1.  The  characteristic  of  the  logarithm  of 
a  number  greater  than  1  in  either  positive  or  zero  and  1  less  than 
'lie  number  of  ditjits  in  the  inteyral  jKtrt  of  the  number. 

2.  The  characteristic  of  the  logarithm  of  a  decimal  is  negative 
and  numerical h/  1  greater  than  the  number  of  ciphers  immediately 
following  the  decimal  point. 

566.  To  avoid  writing  a  negative  characteristic  before  a 
j)Ositive  mantissa,  it  is  customary  to  add  10  or. some  multiple 
of  10  to  the  negative  characteristic,  and  to  indicate  that  the 
number  added  is  to  be  subtracted  from  the  whole  logarithm. 

Thus,  1.6600  is  written  0.(KK)0  -  10 ;  2.3010  is  written  8.3010  -  10  or 
sometimes  18.3010  -  20,  28.:W10  -  30,  etc. 

567.  It  is  evident,  also,  from  the  examples  in  §  564,  that  in 
I  lie  logarithms  of  numbers  expressed  by  the  same  figures  in 
the  same  order,  the  decimal  parts,  or  mantissas,  are  the  same, 
and  the  logarithms  differ  only  in  their  characteristics.  Hence, 
tables  of  logarithms  contain  only  the  mantissas. 

568.  The  table  of  logarithms  on  the  two  following  pages 
^dves  the  decimal  parts,  or  mantissas,  to  the  nearest  fourth 
place,  of  the  common  logarithms  of  all  numbers  from  1  to  1000. 

569 .  To  find  the  logarithm  of  a  number. 

EXERCISES 

1.   Find  the  logarithm  of  765. 

Solution.  —  In  the  following  table,  the  letter  N  designates  a  vertical 
folumn  of  numbers  from  10  to  99  inclusive,  and  also  a  horizontal  row  of 
ti^'ures  0,  1.  2,  3,  4,  6,  6.  7,  8,  9.  The  first  two  figures  of  765  appear  as 
tl)e  miiiiluT  76  in  the  vertical  column  marked  N  on  page  427,  and  the 
tiiird  fiuuie  5  in  the  horizontal  row  marked  N. 

In  the  same  horizontal  row  as  76  are  found  the  mantissas  of  the  loga- 
rithm.s  of  the  numbers  760,  761,  762,  763,  764,  766,  etc.  The  mantissa  of 
the  logarithm  of  765  is  found  in  this  row  under  6,  the  third  figure  of  765. 
It  is  8837  and  means  .8837. 

By  l*rin.  1 ,  the  characteristic  of  the  logarithm  of  765  is  2. 

Hence,  the  logarithm  of  765  is  2.8837. 


426 


L0GA1UTHA18 

Table  of  Common  Logarithms 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

II 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1 106 

13 

1139 

^^3 

1206 

1239 

1271 

1303 

1335 

^3^7 

1399 

1430 

M 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

^^73 

1703 

1732 

15 

1 761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

220I 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

29CX) 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3S38 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4^50 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

501 1 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5"9 

5132 

5145 

5^59 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

55o-^ 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

734.8 

7356 

7364 

7372 

7380 

7388 

7396 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

LOGARITHMS 

Table  df  Common-  Ioi.xriiums 


427 


N 

0      1  1 

2 

3  \    ^ 

5 

6  1  7  ,  8 

9 

55 

7404 

7412 

74»9 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

75»3 

7520 

7528 

7536 

7543 

755' 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7004 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

gf8 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8035 

8041 

8048 

80C5 

64 

8062 

8069 

8075 

8082 

80S9 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

82C2 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

826i 

8267 

8274 

8280 

8287 

8=93 

8299 

8306 

8312 

8319 

68 

8325 
8388 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

!79l 

fZ97 

8802 

76 

8808 

8814 

8820 

8825 

8S31 

8837 

8S42 

8848 

8854 

8859 

77 

8865 

8871 

8S76 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

908^ 

9090 

9096 

9101 

9106 

9112 

9II7 

9122 

9128 

^'11 

82 

9138 

9>43 

9149 

9154 

9«59 

9>65 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

957» 

9576 

^fi 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 
1  9685 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9689 

9694 

9699 

9703 

9708 

97'3 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 
9841 

9800 

9805 

9809 

9814 

9818 

96 

!  9823 

9827 

9832 

9836 

9845 

9850 

9854 

9859 

9863 

97 

1  9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

1  99»2 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

1  9956 

9961 

9965  9969 

9974 
4 

9978 

9983  \   9987 

9991 

9996 

N'  0 

1  :  2  !  3 

5 

6    7    8  !  9 

428  LOGARITHMS 

2.    Find  the  logarithm  of  4. 

Solution.  — Although  the  numbers  in  the  table  appear  to  begin  with 
100,  the  table  really  includes  all  numbers  from  1  to  1000,  since  numbers 
expressed  by  less  than  three  figures  may  be  expressed  by  three  figures  by 
adding  decimal  ciphers.  Since  4  =  4.00,  and  since,  §  567,  the  mantissa 
of  the  logarithm  of  4.00  is  the  same  as  that  of  400,  which  is  .6021,  the 
mantissa  of  the  logarithm  of  4  is  .0021. 

By  Prin.  1,  the  characteristic  of  the  logarithm  of  4  is  0. 

Therefore,  the  logarithm  of  4  is  0.6021. 

Verify  the  following  from  the  table  : 

3.  log  10    =1.0000.  9.  log  .2     =  9.3010 -m 

4.  log  100  =  2.0000.  10.  log  542  =2.7340. 

5.  log  110  =  2.0414.  11.  log  345  =  2.5378. 

6.  log  2      =0.3010.  12.  log  5.07  =  0.7050. 

7.  log  20    =1.3010.  13.  log  78.5  =  1.8949. 

8.  log  200  =  2.3010.  14.  log  .981  =  9.9917  - 10. 

15.    Find  the  logarithm  of  6253. 

Solution.  —  Since  the  table  contains  the  mantissas  not  only  of  the 
logarithms  of  numbers  expressed  by  three  figures,  but  also  of  logarithms 
expressed  by  four  figures  when  the  last  figure  is  0,  the  mantissa  of  the 
logarithm  of  625  is  first  found,  since,  §  567,  it  is  the  same  as  the  mantissa 
of  the  logarithm  of  6250.     It  is  found  to  be  .7959. 

The  next  greater  mantissa  is  .7966,  the  mantissa  of  the  logarithm  of 
6260.  Since  the  numbers  6250  and  6260  differ  by  10,  and  the  mantissas 
of  their  logarithms  differ  by  7  ten-thousandths,  it  may  be  assumed  as 
sufficiently  accurate  that  each  increase  of  1  unit,  as  6250  increases  to 
6260,  produces  a  corresponding  increase  of  .1  of  7  ten-thousandths  in  the 
mantissa  of  the  logarithm.  Consequently,  3  added  to  6250  will  add  .3 
of  7  ten-thousandths,  or  2  ten-thousandths,  to  the  mantissa  of  the  loga- 
rithm of  6250  for  the  mantissa  of  the  logarithm  of  6253. 

Hence,  the  mantissa  of  the  logarithm  of  6253  is  .7959  -1-  .0002,  or  .7961. 

Since  6253  is  an  integer  of  4  digits,  the  characteristic  is  3  (Prin.  1). 

Therefore,  the  logarithm  of  6253  is  3.7961. 

Note.  —  The  difference  between  two  successive  mantissas  in  the  table 
is  called  the  tabular  difference. 


LOGARITHMS  429 

Find  the  logarithm  of : 

16.  1054.                      20.  la.Oy.  24.  .09096. 

17.  1272.                      21.  3.060.  26.  .10126. 

18.  .0166.                     22.  441.1.  26.  54.675. 

19.  1906.                     23.  .7854.  27.  .09886. 

570.  To  find  a  number  whose  logarithm  is  given. 

The  number  that  corresponds  to  a  given  logarithm  is  called 
its  antilogarithm. 

Thus,  since  the  logarithm  of  62  is  1.7924,  the  antilogarithm  of  1.7924 
is  62. 

BXERCISBS 

571.  1.   Find  the  number  whose  logarithm  is  0.9472. 

Solution. — The  two  mantissas  adjacent  to  the  given  mantissa  are 
.9469  and  .9474,  corresponding  to  the  numbers  8.86  and  8.86,  since  the 
given  characteristic  is  0.  The  given  mantissa  is  3  ten-thousandths  greater 
than  the  mantissa  of  the  logarithm  of  8.85,  and  the  mantissa  of  the 
logarithm  of  8.86  is  5  ten-thousandtlis  greater  than  that  of  the  logarithm 
of  8.86. 

Since  the  numbers  8.85  and  8.86  differ  by  1  one-hundredth,  and  the 
mantissas  of  their  logarithms  differ  by  5  ten-thousandths,  it  may  be 
assumed  as  sufficiently  accurate  that  each  increase  of  1  ten-thousandth 
in  the  mantissa  is  produced  by  an  increase  of  |  of  1  one-hundredth  in  the 
number.  Consequently,  an  increase  of  3  ten-thousandths  in  the  njan- 
tissa  is  produced  by  an  increase  of  |  of  1  one-hundredth,  or  .006,  in  the 
number. 

Hence,  the  number  whose  logarithm  is  0.9472  is  8.866. 

2.   Find  the  antilogarithm  of  9.4180  - 10. 

SoLUTioK.  —  Given  mantissa,       .4180 

Mantissa  next  less,  .4106  ;  figures  corresponding,  261. 

Difference,  14 

Tabular  difference,  17)14(.8 

Hence,  the  figures  corresponding  to  the  given  mantissa  are  2618. 
Since  the  characteristic  is  9—  10,  or  —  1,  the  number  is  a  decimal  with 
no  ciphers  immediately  following  the  decimal  point  (Prin.  2). 
Hence,  the  antilogarithm  of  9.4180  -  10  is  .2618. 


430  L()(;akitiims 

Find  the  antilogarithiii  of : 

3.  0.3010.  8.    8.9545.  13.  9.3685-10. 

4.  1.6021.  9.   0.8794.  14.  8.9932-10. 

5.  2.9031.  10.   2.9371.  15.  8.9535-10. 

6.  1.6669.  11.    0.8294.  16.  7.7168-10. 

7.  2.7971.  12.    1.9039.  17.  6.7016-10. 

572.  Multiplication  by  logarithms. 

Since  logarithms  are  the  exponents  of  the  powers  to  which  a 
constant  number  is  to  be  raised,  it  follows  that : 

573.  Prixciple.  —  The  logarithm  of  the  2'>''od>(ct  of  two  or 
mote  numbers  is  equal  to  the  sum  of  their  logarithms ;  that  is, 

To  any  base,         log  {mn)  =  log  m  +  log  n. 

For,  let  logrt  m  =  x  and  log„  n  —  y,  a  being  any  base. 

It  is  to  be  proved  that        loga  (^mn)  —  x  +  y. 


§  559, 

a^  =  m. 

and 

ay  =  n. 

Multiplying,  § 

88, 

gX+y    —    JflYl^ 

Hence,  § 

560, 

log„ 

(mn)  =  x  +  y 

=  logy  m  +  log 

EXERCISES 

574.    1. 

Multiply 

.0381  by  77. 

Solution 

Prin.,  §  573, 

log( 

.0381  X  77) 

=  log  .0381+ log  77. 

log  .0381 

=  8.5809-10 

- 

log  77 

=   1.8865 

Sum  of  logs 

=  10.4674  -  10 

=  0.4674 

1 

0.4674 

=  log  2.934. 

... 

.0381  X  77 

=  2.934. 

LOGARITHMS  431 

NoTR.  —  Three  figures  of  a  number  corn-iiMiKiinir  to  a  logarithm  may 
be  f omul  from  this  table  with  absolute  a<iura(v,  :iiid  in  most  teases  the 
fourth  will  be  correct.  In  finding  logarithms  or  antilogarithms,  allowance 
should  be  made  for  the  figures  after  the  fourth,  whenever  they  express  .6 
or  more  than  .5  of  a  unit  in  the  fourth  place. 

Multiply : 

2.  3.8  by  50.  6.  2.20  by  85.  10.  289  by  .7854. 

3.  72  by  39.  7.  7.25  by  240.  ii.  42.37  by  .236. 

4.  8.5  by  6.2.  8.  3272  by  751  12.  2912  by  .7281. 

5.  1.64  by  35.  d.  .892  by  .805.  13.  1.414  by  2.829. 

575.  Division  by  logarithms. 

Since  the  logarithms  of  two  numhors  to  a  common  base 
represent  exponents  of  the  same  number,  it  follows  that: 

576.  Pkixciplk.  —  The  logarithm  of  the  fjuotieut  of  two  num- 
hern  is  equal  to  the  lofjarifhm  of  the  dividend  minuet  the  logarithm 
of  the  dwiaor  ;  that  is, 

To  any  base,        log  (m  -i-  n)  =  log  m  —  log  n. 
For,  let  logo  m  =  x  and  loga  n  =  y,  a  being  any  base. 

It  is  to  be  proved  that        loga(m  -^  n)  =  ar  -  //. 
§  559,  'I'      III. 

and  a*  =  n. 

Dividinir,  $  127,  a'-'  =  m  ^  n. 

II*  ;  logo  (m  ^ /J  )=•'■— // 

=  ioga  w  -  loga  n. 

EXERCISES 

577.  1.    Divide  .00468  by  73.4. 

Solution 
1  'i  i n. ,  §  570,        log  (.00468  -f-  78.4)  =  log  .00408  -  log  73.4. 
log  .00468  =  7.6702  -  10 

log  73.4    =  1.8657 

Difference  of  logs  =  5.8045  -  10 

5.8046  -  10  =  log  .00006376. 
.-.  .00468  -h  73.4  =  .00006376. 


432  LOGARITHxMS 

2.  Divide  12.4  by  16. 

Solution 

Prin.,  §  576,        log  (12.4  -  16)  =  log  12.4  -  log  16. 
log  12.4  =  1.0934  =  11.0934  -  10 
log  16     =  1.2041 

Difference  of  logs  =    9.8893  -  10 
9.8893  -  10  =  log  .775. 
.-.  12.4  H-  16  =  .775. 

Suggestion.  —  The  positive  part  of  the  logarithm  of  the  dividend  may 
be  made  to  exceed  that  of  the  divisor,  if  necessary  to  avoid  subtracting  a 
larger  number  from  a  smaller  one  as  in  the  above  solution,  by  adding 
10  -  10  or  20  -  20,  etc. 

Divide : 

3.  3025  by  55.  8.  10  by  3.14.  13.  1  by  40. 

4.  4090  by  32.  9.  .6911  by  .7854.  14.  1  by  75. 

5.  3250  by  57.  10.  2.816  by  22.5.  15.  200  by  .5236. 

6.  .2601  by  .68.  11.  4  by  .00521.  16.  300  by  17.32. 

7.  3950  by  .250.  12.  26  by  .06771.  17.  .220  by  .3183. 

578.  Extended  operations  in  multiplication  and  division. 

Though  negative  numbers  have  no  common  logarithms,  opera- 
tions involving  negative  numbers  may  be  performed  by  con- 
sidering only  their  absolute  values  and  then  giving  to  the  result 
the  proper  sign  without  regard  to  the  logarithmic  work. 

Since  dividing  by  a  number  is  equivalent  to  multiplying  by 
its  reciprocal,  for  every  operation  of  division  an  operation  of 
multiplication  may  be  substituted.  In  extended  operations  in 
multiplication  and  division  with  the  aid  of  logarithms,  the 
latter  method  of  dividing  is  the  more  convenient. 

579.  The  logarithm  of  the  reciprocal  of  a  number  is  called 
the  cologarithm  of  the  number. 

The  cologarithm  of  100  is  the  logarithm  of  y^,  which  is  —  2. 
It  is  written,  colog  100  =—2. 


LOGARITHMS  433 

580.  Since  the  logarithm  of  1  is  0  and  the  logarithm  of  a 
(jiiotient  is  obtained  by  subtracting  the  logarithm  of  the  divisor 
t  rum  that  of  the  dividend,  it  is  evident  that  the  cologarithm 
nt  a  number  is  0  minus  the  logarithm  of  the  number,  or  the 
l()f,'arithm  of  the  number  with  the  sign  of  the  logarithm 
(hanged ;  that  is,  if  log„  m  =  x,  then,  colog.  m  =  —  x. 

Since  subtracting  a  number  is  equivalent  to  adding  it  with 
its  sign  changed,  it  follows  that: 

581.  Principle.  —  Instead  of  mhtracting  the  logarithm  of  the 
(in-isor  from  thoJt  of  the  dividend^  the  cologarithm  of  the  divisor 
may  be  added  to  the  logarithm  of  the  dividend;  that  is, 

To  any  base,         log  {m  -h  n)  =  log  w  +  colog  w. 

BXERCISBS 

582.  1.   Find  the  value  of  '^^A^f  ^!f^  by  logarithms. 

468  X  16.6  X  .029     -^      ^ 

Solution 

:««3x58A2LM=  .063  X  58.6  X  799  X  J- X  ^  X  J-. 
468  X  16.6  X  .029  468      15.0      .029 

log  .063  =  8.7903  -  10 

log  68.6=  1.7672 

log  799=  2.9026 

colog  468=  7.3391  -10 

colog  16.6=  8.8069-10 

colog  .029=  1.6376 


log  of  result  =  31.1626  —  80 
=    1.1626. 
.-.  result  =  14.21. 
Find  the  value  of : 
^     110  X  3.1  X  .653  ,       16  X. 37x26.16 


33  X  7.854  X  1.7  11  x  8  x  .18  x  6.67 

MILNE'8   8TAV1V     M.r,    — 28 


434  LOGARITHMS 


(-3.04)  X  .2608  ^    .4051  x  (-  12.45) 

^'     2.046  X  .06219  '  '   /      or.oox..   .v-.... 


600  X  5  X  29 


5.   ^ 


6. 


.7854  X  25000  x  81.7 
3.516  x  485x65 


9. 


(-  8.988) 

X  .01441^ 

78  X  52  X 

1605 

338  X  767 

X  431 

.5  X  .315 

X428 

3.33  X  17  X  18  X  73  .317  x  .973  x  43.' 


583.  Involution  by  logarithms. 

Since  logarithms  are  simply  exponents,  it  follows  that : 

584.  Principle.  —  The  logarithm  of  a  power  of  a  number  is 
equal  to  the  logarithm  of  the  number  multiplied  by  the  index  of  the 
power ;  that  is, 

To  any  base,  log  771"  =  n  log  m. 

For,  let  logrt  m  =  x,  and  let  n  be  any  number,  a  being  any  base. 

It  is  to  be  proved  that  logo  m"  =  nx. 

§  559,  a^  =  m. 

Raising  each  member  to  the  nth  power,  Ax.  6  and  §  276,  2, 

Hence,  §  560,  loga  wi«  =  nx  =  n  loga  m. 

EXERCISES 

585.  1.    Find  the  value  of  .25-. 

SoLUTioy 
Prin.,  §  584,  log  .252  ^  2  log  .25. 

log  .25=    9.3979-10. 
2  log  .25  =  18.7958  -  20 
=    8.7958  -  10. 
8.7958-  10  =  log  .06249. 
.-.  .25-^  =  .06249. 

Note.  —By  actual  multiplication  it  is  found  that  .252=  .0625,  whereas 
the  result  obtained  by  the  use  of  the  table  is  .06249. 

Also,  by  multiplication,  IS^  =  324,  whereas  by  the  use  of  the  table  it  is 
found  to  be  324.1.  Such  inaccuracies  must  be  expected  when  a  four-place 
table  is  used. 


LOGARITHMS  435 

Find  by  logarithms  the  value  of : 

2.  7\  7.     .78«.  12.    4.071  17.  (^y, 

3.  IV.  8.    8.052.  13      5438  18.  (|)«. 

4.  (-47)1     9.    8.33^  14.    (-7/.  19.  (^\\%)\ 

5.  4.92.  10.    6.6P.  15.    1.02^  20.  (^^y. 

6.  r).l>-'.  11.    .714«.  16.    1.738^  21.  {^^sV. 

586.  Evolution  by  logarithms. 

Since  logarithms  are  simply  exponents,  it  follows  that : 

587.  Princi  PLE.  —  The  Ifjyarithm  of  a  root  of  a  u  umber  is  equal 

in  the  hxjarithiu  of  the  munber  divided  by  the  index  oftfie  required 

root;  that  is, 

To  any  base,  log  ^m  =  l^ii!.*. 

n 

For,  let  log,,  m  =  x  and  let  n  Ik*,  any  number,  a  being  any  base. 
It  is  to  be  proved  that      log,,  Vm  =  x  -■-  n. 
§  559,  (i^  —  III. 

Taking  the  nth  root  of  each  member,  Ax.  7  and  §  290, 

Hence,  §  560,  log«  ^m  =  x-^n  =  !^??-^\ 

n 

EXERCISES 

588.  1.    Find  the  square  root  of  .1296  by  logarithms. 

Solution 

I'liii.,  §  587,  log>/.T29«  =  J  log  .1296. 

log.  1296  =  9.1126-10. 

2)19.1126-20 
9.506.J  -  10 

9.556.3  -  10  =  log  ..360. 

.-.  Vim  :=  .30. 


436  LOGARITHMS 

Find  by  logarithms  the  value  of : 

2.  225i  8.    (-1331)1 

3.  12.25^.  9.  1024^. 

4.  .20232.  10.  .6724^. 

5.  326i  11.  5.929i 

6.  .512I  12.  .4624^ 

7.  .1182^  13.  1.464li 

Simplify  the  following : 

26  1^^  31     14.5^^:T6 

15x3.1416  '  11 


14. 

V2. 

20. 

</-2. 

15. 

V3. 

21. 

^.027. 

16. 

V5. 

22. 

V304. 

17. 

V6. 

23. 

V.90. 

18. 

^2. 

24. 

v^>. 

19. 

VI. 

25. 

-V/.032. 

27.         (-^Q^)^     .  32. 

48  X  64  X  11 


U34  X  96^ 
^'64xl^)00' 


2g     52^  X  300  _^  33      .32x5000x18 


12  X  .31225  X  400000  3.14  x  .1222  x  8 

4: 


29.       /        400  34     11x2.63x4.263 


55  X  3.1416  48  x  3.263 


30    50  x^''  q^    J    ^^Q^ 

'''•  ^^""s^  ^'-  V(3i:06/- 

36.    22  X  (i)^  X  v'^  X  VX 

37.  Applying  the  formula  A  =  ttt^,  find  the  area  (A)  of  a 
circle  whose  radius  (r)  is  12.35  meters,     (tt  =  3.1416.) 

38.  Applying  the  formula  r=  |  -n-r'^,  find  the  volume  (F)  of 
a  sphere  whose  radius  (r)  is  40.11  centimeters. 

39.  The  formula  V=  .7854  dH  gives  the  volume  of  a  right 
cylinder  d  units  in  diameter  and  I  units  long,  F,  d,  and  Z  being 
corresponding  units.  How  many  feet  of  No.  00  wire,  which 
has  a  diameter  of  .3648  inches,  can  be  made  from  a  cubic  foot 
of  copper  ? 


LOGARITHMS  437 

589.  Solution  of  exponential  equations. 

Exponential  equations,  or  equations  that  involve  unknown 
exponents,  are  solved  by  the  aid  of  the  principle  that,  in  any 
system,  eqatd  numbers  have  equal  logarithms. 

In  simple  cases  the  solution  of  such  equations  may  be  per- 
iormed  by  inspection,  but  in  general  it  is  necessary  to  use  a 
table  of  logarithms. 

EXERCISES 

590.  1.    Find  the  value  of  a;  in  the  equation  2'  =  32V2. 

SOLUTIOX 

2«  =  32V2=2»2^=2^; 

th(  K  tuK  .  log  (2*)  =  log  (2  2), 

or,  §084,  jclog2=  Vlog2. 

Dividing  by  log  2,  x^^. 

2.  Find  the  value  of  x  in  the  equation  2*  =  48. 

Solution 
Taking  the  logarithm  of  each  member, 

X  log  2  =  log  48. 

log   2 

=  i:«^  =  5.69-. 
0.3010 

3.  Solve  the  equation  3**  -  20  •  .r  +  99  =  0  for  x. 

Solution 
Factoring  the  given  equation, 

(3«_9)(3»-ll)  =  0. 

.-.  3«  =  9orll. 

Solving  the  equation  3*  =  9  by  inspection,  since  9  =  3*, 

a;=2. 

Taking  the  logarithm  of  each  member  of  H'  =  11. 

X  log  3  =  log  1 1 . 

.^^l^il^lJ>4L4^2.18H.. 
log  3     0.4771 

Therefore,  the  value  of  x  is  either  2  or  2.18+. 


488 


LOGARITHMS 


4.    Given  x^  =  y^  and  x^  =  if,  to  find  x  and  y. 

Solution. — Raising  the  members  of  tiie   first  equation   to  the  xt\\ 
power,  and  those  of  the  second  equation  to  the  8d  power, 


and 

x^y  =  y^^. 

Hence,  by  inspection, 

2x  =  Sy. 

Squaring,  since  4 

a;-  = 

Ay^ 

4x^  =  9  ?/2  =  4 
.•.y  =  Oorf, 

2,3. 

and 

5C  =  0  or  V-. 

5.    Given  3*  = 

2y  and  2 

""  =  y,  to  find  x 

and 

y- 

Solution. 

S-=2y. 

2^  =  y- 

0) 

(2) 

Dividing  (1)  by  (2), 

(1.5)^  =  2. 

.-.  X 

log  1.5  =  log  2. 

Hence,  by  tables. 

log  2 
^-  log  1.5  " 

0.3010 
"0.1761 

(3) 

By  logarithms, 

log  a;  =  0.2328; 

(4) 

whence,  by  tables, 

x=  1.709. 

(5) 

From  (2), 

logy  =  x  log  2. 

Then, 

log  log  y  =  log  ic  +  log  log  2 

by  (4)  and  tables, 

=  0.2828  +  1.4786^ 

=  1.7114. 

Hence,  by  tables. 

log  y  =  0.5145  ; 

whence, 

y  =  8.270. 

Solve  the  following : 

6.  3^  =  81. 

7.  4'  =  10. 

12. 
13. 

2'^ 
(2- 

=  512. 
y  =  256. 

17. 

[2^+1 

=  6, 

=  3^ 

8.  2^=80. 

9.  3^'=9^ 

14. 

|3^ 

14 

'  =  2y, 
^  =  20y. 

18. 

12^-^ 

=  32, 
=  4. 

10.  2=^' =512. 

11.  5^' =625. 

15. 
16. 

32. 
lo" 

+  243  =  36-3^ 

■  ]os^x  =  log  2. 

19. 

f2^  = 

[x  = 

yi 

1  +  log  y. 

LOGARITHMS  439 

591.  Logarithms  applied  to  the  solution  of  problems  in  com- 
pound interest  and  annuities. 

SiiHje  the  amount  of  any  principal  at  6  %  interest,  compounded 
annually,  for  1  year  is  1.06  times  the  principal;  for  two  years, 
l.Of)  X  l.CK),  or  1.06^,  times  the  principal;  for  8  years,  1.06  x 
1.0()  X  1.06,  or  1.0()^,  times  the  principal,  etc.,  the  amount  (A) 
>  •!'  any  principal  {P)  for  n  years  at  any  rate  per  cent  (r)  will  be 

Expressing  this  formula  by  logarithms, 

log  A  =  log  P  +  n  log  (1  -f  r).  (1) 

.-.  log  P=  log  A-n  log(l  -h  r) ;  (2) 

also  log(l-fr)=^^--^-^^g^:  (3) 

n 

and  n  =  '^fA^^,  (4) 

log(l+r) 

EXERCISES 

592.  1.    What  is  the  amount  of  S475  for  10  years  at  6% 

impound  interest? 

Soli  i  Jo.N 

log476  =  2.6767 
log  1.06W  =  0.2580 
log^        =2.9297 
.-.^=$860.60. 

N<»TK.  —  III  accordance  with  the  note  on  page  4;^1.  antilogarithm.s  are 
carried  out  only  to  the  nearest  fourth  significant  figure. 

Find  the  amount,  at  compound  interest,  of : 

2.  $ 225,  5  years,  8  %.  4.    $ 400,  10  years,  3  %. 

3.  $  700,  5  years,  6  ^c  5-    $  1200,  20  years,  4  %. 


440  LOGARITHMS 

6.  What  principal  will  amount  to  $  1000  in  10  years  at  5  % 
compound  interest  ? 

7.  What  sum  of  money  invested  at  4  %  compound  interest, 
payable  semiannually,  will  amount  to  {^743  in  10  years? 

8.  What  principal  loaned  at  4%  compound  interest  will 
amount  to  f  loOO  in  10  years? 

9.  What  sum  invested  at  4  %  compound  interest  at  a  child's 
birth  will  amount  to  $  1000  when  he  is  21  years  old  ? 

10.  In  what  time  will  $800  amount  to  $1834.50,  if  put  at 
compound  interest  at  5  %  ? 

11.  What  is  the  rate  per  cent  when  $300  loaned  at  com- 
pound interest  for  6  years  amounts  to  $402? 

12.  A  man  agreed  to  loan  $1000  at  6%  compound  interest 
for  a  time  long  enough  for  the  principal  to  double  itself. 
How  long  was  the  money  at  interest  ? 

593.  A  sum  of  money  to  be  paid  periodically  for  a  given 
number  of  years,  during  the  life  of  a  person,  or  forever,  is 
called  an  annuity. 

The  payments  may  be  made  once  a  year,  or  twice,  or  four 
times  a  year,  etc. 

Interest  is  allowed  upon  deferred  payments. 

594.  To  find  the  amount  of  an  annuity  left  unpaid  for  a  given 
number  of  years,  compound  interest  being  allowed. 

An  annuity  of  a  dollars  per  year,  payable  at  the  end  of  each 
year,  will  amount  to  a  dollars  at  the  end  of  the  first  year.  If 
unpaid  and  drawing  compound  interest  at  a  rate  r,  the  accumu- 
lation at  the  end  of  the  second  year  will  be  a-\-a{l  +  r)  dol- 
lars; at  the  end  of  the  third  year,  a -f- a(l  +  r) -|- a(l  +  r)- 
dollars;  and  so  on. 

Let  a  represent  the  annuity,  n  the  number  of  years,  r  the 
rate,  and  A  the  whole  amount  due  at  the  end  of  the  7ith.  year. 

Then,    ^  =  « +  a(l  +  r)  +  a(l +  ?•)-+ •••  +  a(l4- ^T"^ 


LOGARITHMS  441 

Since  the  terms  of  ^1  form  a   geometrical  progression   in 
which  1  -f  /•  is  the  mtio,  §  525,  the  sum  of  the  series  is 

^  =  ^[(l  +  r)>-l]. 

EXERCISES 

595.    1.  What  will  be  the  amount  of  annuity  of  $10()  re- 
maining unpaid  for  10  years  at  6%  compound  interest? 

Solution 


and 


A^ 

".[(l+r) 

«-!]. 

log   1.06W  = 

.2680 

.-.  1.0610  = 

1.7904 

1.06W  -  1  = 

.7904 

log  100  = 

2.0000 

log. 7904  = 

9.8978  - 

10 

colog  .06  = 

1.2218 

.•.log^  = 

13.1196- 

10 

= 

3.1196. 

A  = 

$1317,  the  amount  of  the  annuity. 

Hence, 

2.  To  what  sum  will  an  annuity  of  $  26  amount  in  20  years 
Mt  4%  compound  interest? 

3.  What  will  be  the  amount  of  an  annuity  of  $17.76  re- 
maining unpaid  for  25  years,  at  3J%  compound  interest  ? 

4.  What  annuity  will  amount  to  1^1000  in  10  years  at  5% 
'  t  impound  interest  ? 

5.  What  annuity  will  amount  to  $6000  in  12  years  at  3^^ 
impound  interest? 

596.  A  sum  that  will  amount  to  the  value  of  an  annuity, 
if  put  at  interest  at  the  given  rate  for  the  given  time,  is  called 
the  present  value  of  the  annuity. 

Sometimes  annuities,  drawing  interest,  are  not  payable  until  after  a 
I  tain  number  of  years. 


442  LOGARITHMS 

597.  Let  P  denote  the  present  value  of  an  annuity  due  in  n 
years,  with  compound  interest  at  a  rate  r.  Then,  the  amount 
of  P  at  the  end  of  the  period  will  be  found  thus  : 

By  §  591,  A=P(l-\-ry\ 

But,  §  594,  ^  =  ^ [(l-f  ry -  1]. 

Hence,  Ax.  5,       P(l  +  rf  =  -[(1  +  rf  -  1]. 

r*     (1  +  r)" 

EXERCISES 

598.  1.  What  is  the  present  value  of  an  annuity  of  $100 
to  continue  10  years  at  6  %  compound  interest  ? 


SOLUT 


I  ox 


P 


«.(!  -f  ry 


(1  +  ry 
logl.06i«=      .2580 
.-.1.0610=    1.7904 

and  1.0610-1=      .7904 


log  100  =  2.0000 

log  .7904  =  9.8978  -  10 

colog  .06  =  1.2218 

coloo- 1.0610  =  9.7470  -  10 


.-.  log  P=  22.8666 -20 
=    2.8666. 
Hence,  P=t  736.50,  the  present  value. 

2.  What  is  the  present  value  of  an  annuity  of  $300  for 
5  years  at  4  %  compound  interest  ? 

3.  What  is  the  present  value  of  an  annuity  of  $  1000  to  con- 
tinue 20  years,  if  compound  interest  at  41  %  is  allowed  ? 

4.  Find  the  present  value  of  an  annuity  of  £  2000  payable 
in  10  years,  interest  being  reckoned  at  3  %. 


PERMUTATIONS  AND  COMBINATIONS 


599.  All  the  different  orders  in  which  it  is  possible  to 
xrruiKje  a  given  number  of  things,  by  taking  either  some  or  all 
of  them  at  a  time,  are  called  the  permutations  of  the  things. 

Thus,  the  permutations  of  the  letters  a  and  b  are  a6,  6a  ;  the  permuta- 
lions  of  three  letters  a,  6,  and  c,  two  at  a  time,  are  afc,  ac,  ha,  be,  ca,  cb. 

600.  All  the  different  selections  that  it  is  possible  to  make 
t  lom  a  given  number  of  things,  by  taking  either  some  or  all  of 
them  at  a  time,  without  regard  to  tlie  order  in  which  they  are 
placed,  are  called  the  combinations  of  the  things. 

Thus,  while  the  permutations  of  three  letters,  a,  6,  and  c,  two  at  a  time, 
are  ah  and  ba,  be  and  c6,  and  ca  and  ac^  their  combinations,  two  at  a 
time,  are  ab  (or  ba,  but  not  both),  be  (or  eft),  and  ar  (or  ca)  ;  again,  the 
six  permutations  of  these  three  letters  among  themselves,  viz.,  abc,  neb, 
'■'-a,  bac,  cab,  and  cba,  form  but  one  combination,  abc  (or  acb,  or  bca,  or 
'r,  or  c<ib,  or  cba). 

It  is  evident  that  there  can  be  only  one  combination  of  any  number  of 
tliiii::s  taken  all  at  a  time. 

601.  Notation.  —  The  symbol  for  the  number  of  pemiutations 
(.1  n  different  things,  taken  r  at  a  time,  is  P^;  of  n  different 
things,  taken  n  at  a  time,  or  all  together,  is  P*. 

Instead  of  P^,  sometimes  "P^,  ^P^,  or  P^  ^  is  used.    Similarly,  for  P;, 
luetimes  "P^,  ^P^,  or  P^  ^  is  used. 

The  symbol  for  the  number  of  combinations  of  n  different 
things,  taken  r  at  a  time,  is  C?;  of  n  different  things,  taken  n 
at  a  time,  or  all  together,  is  Cj. 

Instead  of  C^,  sometimes  *C,  ^C^,  or  C^^  is  used.  Similarly,  for  C*, 
sometimes  "C  ,    C  ,  or  C      is  used. 

443 


444  PKKMUTATIONS   AND   COMP,I  .\  A TTONS 

602.  Tlie  product  of  the  successive  integers  from  1  to  n,  or 
from  n  to  1,  inclusive,  is  called  factorial  n,  written  \n,  or  n\. 

1 5  =  1  X  2  X  8  X  4  X  •'),  or  5  X  4  X  3  X  2  x  1 ; 

j  w  =  1  •  2  .  8  •■•  (»  -  2)  {n  -  1)«,  or  n{n  -  1)  {n  -  2)  (n  -  8)  .-  8  •  2  •  1. 

603.  To  find  the  number  of  permutations  of  n  different  things 
taken  r  at  a  time. 

Since  the  permutations  of  a,  b,  and  c,  taken  2  at  a  tiuie,  are  ah 
and  ac,  ha  and  he,  ca  and  ch,  formed  by  writing  after  each  of  the 
letters,  a,  h,  and  e,  each  of  the  other  letters  in  turn,  the  number 
of  permutations  of  3  different  things  taken  2  at  a  time  is  3  x  2. 

The  number  of  permutations  of  n  letters  taken  2  at  a  time 
may  be  found  by  associating  with  each  of  the  n  letters  each  of 
the  n  —  1  other  letters.  Consequently,  the  number  of  permu- 
tations of  n  different  things  taken  2  at  a  time  is  n{n  —  1). 

Since  the  number  of  permutations  of  n  letters  2  at  a  time  is 
nin  —  1),  if  the  letters  are  taken  3  at  a  time  there  will  be  n  —  2 
letters,  each  of  which  may  be  associated  with  each  of  the 
n(ii  —  1)  permutations  of  letters  taken  2  at  a  time.  Hence, 
the  number  of  permutations  of  n  different  things  taken  3  at  a 
*^^^is  n(n-l){n-2). 

Principle  1.  —  The  numher  of  jjermutatio^is  of  n  different 
tlmigs  taken  r  at  a  time  is  equal  to  the  continued  product  of  the 
natural  numbers  from  n  to  7i—(r—l)  incht'Sive.  TJie  number 
of  factors  is  r.     That  is, 

P!J  =  n(n  —  l)(n  —  2)"'tor  factors 

=  n{n-\)(n-2)-"{n-r  +  l).  (I) 

Multiplying  and  dividing  the  second  member  of  (I)  by 
(n  —  r)(yi  —  r  —  l)(7i  —  r  —  2)  •  •  •  2  •  1 ;  that  is,  by  '\ti  —  r, 

p;=-i^.  (II) 

\n  —  r 

Note.  —  It  will  usually  be  more  convenient  to  employ  formula  (I)  in 
solving  numerical  exercises  ;  but  when  literal  results  are  desired,  formula 
(II)  will  be  preferable. 


PERMUTATIONS   AND   COMBINATIONS  445 

604.  When  /•  =  n,  that  is,  when  the  things  are  taken  all 
.ether,  the  last,  or  nth,  i'M-Xov  in  (I)  is  1.     Consequently, 

Tkixciplk  2. —  Tile  nnnther  of  perniutatious  of  n  differeiU 
tliiiajs  taken  all  at  a  time  in  equal  to  [n.     That  is, 

i>:  =  ui^n  -  l)(n  -  2)  .  •  .  2  . 1  =[n.  (Ill) 

BXSRCISBS 

605.  1.   Three  boys  enter  a  car  in  which  there  are  5  empty 
its.     In  how  many  ways  may  they  choose  seats  ? 
Solution.  —  Since  the  first  boy  may  choose  any  one  of  6  seats,  and 

since  for  each  seat  that  he  may  choose  the  second  boy  may  choose  any 
one  of  the  4  seats  remaining,  the  greatest  ix)ssible  number  of  ways  in 
which  two  of  the  boys  may  be  seated  is  5  x  4.  ^ 

Again,  since  after  each  choice  of  seats  made  by  two  of  the  boys  there 
will  be  left  to  the  third  boy  a  choice  of  one  of  the  3  seats  remaining,  the 
number  of  ways  in  which  all  may  choose  seats  is  5  x  4  x  3,  or  60. 

Or,  by  (I),       P:  =  n{n  -  l)(n  _  2)  •  • .  («  -  r  -f  1)  ; 

that  is,  P|  =  6x4x8  =  60. 

2.  How  many  numbers  between  100  and  1000  can  be  ex- 
pressed by  the  figures  1,  3,  5? 

Solution.  —  Since  the  numbers  lie  between  100  and  1000,  each  must 
be  expressed  by  three  figures.  Hence,  the  number  of  numbers  between 
100  and  KKM)  that  can  be  expressed  by  the  figures  1,  3,  and  5  is  the  same 
ns  the  number  of  permutations  of  these  3  figures  taken  3  at  a  time. 

Since,  Prin.  2,  P|  =[3  =  32  •  1  =  6, 

lit'  are  six  such  numbers.     They  are  185,  158,  351.  315,  513,  and  531. 

3.  How  many  permutations  can  be  made  of  the  letters  in 
the  word  Albany,  each  beginning  with  capital  A  ? 

Solution.  —  Since  A  is  to  be  prefixed  to  each  permutation  of  the  6 
other  letters,  the  required  number  is 

P|  =5x4x3x2x1  =  120. 

4.  In  how  many  orders  may  4  persons  sit  on  a  bench  ? 

5.  How  many  permutations  may  be  made  of  the  letters  in 
the  word  number  ? 


446  PERMUTATIONS   AND   COMBINATIONS 

6.  If  10  athletes  run  a  race,  in  liow  many  ways  may  tlie 
first  and  second  prizes  be  awarded  ? 

7.  In  how  many  different  orders  may  the  colors  violet, 
indigo,  blue,  green,  yellow,  orange,  and  red  be  arranged  ? 

8.  There  are  5  routes  to  the  top  of  a  mountain.  In  how 
many  ways  may  a  person  go  up  and  return  by  a  different  way  ? 

606.  To  find  the  number  of  combinations  of  n  different  things 
taken  r  at  a  time. 

Since  two  letters,  as  a  and  6,  have  two  permutations,  ah  and 
ha,  but  form  only  one  combination,  the  number  of  combina- 
tions of  n  letters  taken  2  at  a  time  is  one  half  the  number  of 
|)ermutations  of  n  letters  taken  2  at  a  time. 

Since  three  letters  taken  3  at  a  time  have  3x2  permuta- 
tions, but  form  only  one  combination,  the  number  of  combina- 
tions of  n  letters  taken  3  at  a  time  is  obtained  by  dividing  the 
number  of  permutations  of  n  letters  taken  3  at  a  time  by  3x2. 

Since  four  letters  taken  4  at  a  time  have  |4  permutations 
but  form  only  one  combination,  to  obtain  the  number  of  com- 
binations of  n  letters  taken  4  at  a  time,  the  number  of  permu- 
tations of  n  letters  taken  4  at  a  time  must  be  divided  by  |4. 
Hence, 

Principle  3.  —  The  number  of  comhinations  of  n  different 
things  taken  r  at  a  time  is  equal  to  the  numher  of  permutations  of 
n  different  things  taken  r  at  a  time,  divided  hy  the  numher  of 
permutations  of  r  different  things  taken  all  together.     That  is, 

pn  _  pn  _^  pr  _  y^(n  —  l)(n  —  2)  '"  to  r  factors 
r{r  —  l)(r  — 2)  .••  to  r  factors 


^ n{n  -l)(n-2)  '"  (n  -  r -f  1) 
~  1.2.3...r 

Or,  by  (II)  and  (III), 

I  n  —  r     ~ 

\r'.)i  —  r 


(IV) 


(V) 


PERMUTATIONS   AND  COMBINATIONS  447 

607.  Since  for  every  combination  ot  /  1  liings  out  of  n  differ- 
ent tilings  there  is  left  a  combination  oi  u  -  /•  things,  it  follows 
that: 

Principle  4.  —  The  number  of  combinations  of  n  different 
filings  is  the  same  ivhen  taken  n  —  r  at  a  time  as  when  taken  r  at 
(I  time.     That  is,  , 

(-.,  =  C^  =  -J2_.  (VI) 

\r\n  —  r 


The  above  principle  may  be  established  as  follows : 


By  (V), 

0=      ^ 

^     |r|n-r 

Substituting  n  - 

-  r  for  r, 

,..     _               ti? 

"  *"     |n  —  r|n  —  (w  - 

j:! 

_     Hi 

\n-r\r 

(1) 


(2) 


Since  the  second  members  of  (1)  and  (2)  are  identical,  Clt_r  =  C^. 
The  above  principle  is  useful  in  abridging  numerical  computations. 
Thus,  the  number  of  combinations  of  18  things  taken  16  at  a  time  is 
computed  by  Prin.  8  as  follows: 

C8  =  I»n.  16 -15.  14.  13-  12.  11     10- 9.8-7    H.  5. 4.  3  ^  ^^ 
^«        1.2. 8.  4.  5.  6.  7.  8.  9.  10.  11.  12. 13. 14.  16. 16 

Hut  by  Prin.  4,  the  computation  is  abridged  as  follows: 

CJ«  =  (718  =  ^^-^  =  158. 
"        -         1-2 


BXBRCISBS 

608.    1.    A  man  has  6  friends  and  wishes  to  invite  4  of  them 
to  dinner.     In  how  many  ways  may  he  select  his  guests  *.' 

SoLUTio!*.  —  Since  each  party,  or  combination,  of  4  guests  could  be 
arranged,  or  permuted,  in  [4  ways,  the  number  of  combinations  must  be 
of  the  number  of  permutations  of  6  things  taken  4  at  a  time. 

Hence,  the  number  of  ways  is 

C^=  P^,-^  P*  =  ^  X  •'^  X  4  y_3  _.  ^  ;^ 
4  4  *       1  X  2    • 


448  PERMUTATIONS   AND   COMBINATIONS 

2.  A  man  and  his  wife  wish  to  invite  11  of  their  friends, 
6  men  and  5  women,  to  dinner,  but  find  that  they  can  enter- 
tain only  8  guests.  In  how  many  ways  may  they  invite  4  men 
and  4  women  ? 

Solution.  — As  in  the  previous  exercise,  4  men  may  be  selected  from 
6  men  in  15  ways,  and  in  a  similar  manner  4  women  may  be  selected 
from  5  women  in  5  ways. 

Since,  when  any  set  of  4  men  has  been  invited,  the  party  of  8  may  be 
completed  by  inviting  any  one  of  5  sets  of  4  women,  the  whole  number  of 
different  parties  that  it  is  possible  to  invite  is  16  x  5,  or  75.     That  is, 

^H^^5^6.5.4.3^5.4.3.2^,,^ 
1.2.3.4      1.2.3.4 

3.  In  how  many  ways  may  a  baseball  nine  be  selected  from 
12  candidates  ? 

4.  How  many  different  combinations  of  5  cards  may  be 
formed  from  52  cards  ? 

5.  Which  is  the  greater,  C'^  or  (7\"?  C'^  or  C'9?  O?  or  C'i? 

6.  From  11  Eepublicans  and  10  Democrats  how  many  dif- 
ferent committees  may  be  selected  composed  of  6  Republicans 
and  5  Democrats  ? 

7.  A  man  forgets  the  combination  of  figures  and  letters  by 
which  his  safe  is  opened.  They  are  arranged  on  the  circum- 
ferences of  three  wheels,  one  bearing  the  numbers  0  to  9  inclu- 
sive, another  the  letters  A  to  M  inclusive,  and  the  third  the 
letters  N  to  Z  inclusive.  What  is  the  greatest  number  of  trials 
he  may  have  to  make  to  open  the  safe  ? 

8.  From  6  consonants  and  4  vowels  how  many  words  may 
be  formed  each  consisting  of  4  consonants  and  2  vowels,  if  any 
arrangement  of  the  letters  is  considered  a  word  ? 

Suggestion.  — The  number  of  combinations  is  04  x  C2;  and  since  by 
permuting  the  letters  of  each  combination  |J>  words  can  be  formed,  the 
number  of  words  is  C4  x  O*  x  |^. 

9.  In  an  omnibus  that  will  seat  8  persons  on  a  side  there 
are  seated  4  persons,  3  on  one  side  and  1  on  the  other.  In  how 
many  ways  may  12  more  persons  be  seated? 


\ 

PKKMLTATIONS   AND   COMBINATIONS  449 

Solution.  —  Since  6  persons  must  take  seats  on  one  side  and  7  persons 
on  the  other,  12  persons  are  to  l)e  divided  into  two  classes,  6  and  7.  The 
number  of  these  combinations,  formula  (V),  is 

Since  each  combination  of  6  may  have  [6  permutations  of  the  5  that 
compose  it,  and  each  combination  of  7  may  have  \J_  permutations  each  of 
which  may  he  associated  with  each  of  the  [6  permutations,  the  required 
number  of  ways  is  C*/  x  P*  x  P? ;  that  is, 

112 

Or,  12  persons  may  be  seated  in  12  seats  in  p1|  =  [12  ways. 

10.  Out  of  20  consonants  and  6  vowels  how  many  words 
containing  3  consonants  and  3  vowels  may  be  formed,  if  any 
arrangement  of  the  letters  is  considered  a  word  ? 

11.  How  many  different  sums  may  be  paid  with  a  cent,  a 
r^itent  piece,  a  dime,  a  quarter,  and  a  dollar  ? 

12.  From  5  boys  and  5  girls  how  many  committees  of  6  may 
be  selected  so  as  to  contain  at  least  2  boys  ? 

13.  If  C;  =  2  Cl  find  the  number  of  things. 

Solution.  —By  formula  (V),  C«  =  — -^^^ —  and  Cj  =  . — — 

|6|«  —  6  |2!n  — 2 


Since  C'5  =  2C". 


[n  2[n 


|6|n-6     |2|n-2 

1        _      1 
|5|n-6     \n-2 

.-.      |n-2  =  |6|n-5. 

In -2 

=  [5  =  6x  4x3x2x1 


n-6 


rl.atis,  (n-2)(n-3)(n-4)  =  6xr,  x  4. 

.-.  n  =  8. 

14.    If  3  C?  =  2  (7*|S  find  7i,  C%  and  0*t^ 
milne's  stand,  alo.  —  21* 


450  PERMUTATIONS  ANJ)  COMBINATIONS 

609.  To  find  the  number  of  permutations  of  n  things  taken  n  at 
a  time  when  they  are  not  all  different. 

If,  in  the  permutation  (a,  b,  c,  d,  e,  f,  g),  the  letters  b,  d,  and 
g  are  permuted  while  the  other  letters  remain  fixed  in  position, 
the  resulting  number  of  permutations  will  be  the  same  as  the 
number  of  permutations  of  b,  cl,  and  g.  If  b,  d,  and  g  are  dif- 
ferent things,  the  number  of  permutations  resulting  will  be  |3^; 
but  if  b,  d,  and  g  become  alike,  there  will  be  but  1  permutation. 

That  is,  the  number  of  permutations  of  any  number  of  things, 
3  of  which  are  alike,  is  equal  to  the  number  of  permutations 
of  the  things,  considered  as  all  different,  divided  by  |3;  if  4  of 
the  things  are  alike,  by  [^;  if  2^  of  the  things  are  alike,  by  \p^ 

Hence,  it  follows  that : 

Principle  5.  —  The  number  of  permutations  of  n  things,  taken 

\n 
all  together,  when  p  of  them  are  alike,  is  _. 

[P 
If  q  of  the  remaining  7i—p  different  things  become  alike, 
but  different  from  the  p  like  things,  the  number  of  permuta- 
tions must  be  divided  by  [qj  if  r  others  become  alike,  by  [r; 
etc.     Hence,  it  follows  that : 

Principle  6.  —  The  number  of  permutations  of  n  things,  taken 

all  together,  tvhen  p  of  them  are  of  one  kind,  q  of  another,  r  of 

.     ■     1^ 
another,  etc.,  is        — 


EXERCISES 

610.  1.  How  many  permutations  may  be  made  with  the 
letters  of  the  word  Mississippi  taken  all  together  ? 

Ill 
Solution.  —  The  number  is  ,   —    =  34650. 
[41412 

2.  How  many  permutations  may  be  made  with  the  letters 
of  each  of  the  following  words,  taken  all  at  a  time  in  each 
case:  characteristic,  coefficient,  ecclesiastical,  divisibility? 

3.  How  many  permutations  may  be  made  with  the  letters 
represented  in  the  product  a'^b^c-  written  out  in  full  ? 


PKKMUTATIONS   AXD  COMBINATIONS  451 

611.  To  find  the  total  number  of  combinations  of  n  different 
things. 

The  nuuiber  of  combinations  of  n  different  things  taken  suc- 
.  .'ssively  1,  2,  3,  •••  n  at  a  time  is  called  the  total  number  of 

iiibinations  of  n  things. 

The  total  number  of  combinations  of  2  things  is 

C'i'-f  Ci  =  2-f  1=8,  or2«-l. 

The  total  number  of  combinations  of  3  things  is 
Cf+C|-fCi  =  34-3-hl=7,  or2»-l. 

The  total  number  of  combinations  of  4  things  is 

<7t+C'J-fC.1-fCJ  =  4 +6+4 -hi  =15,  01-2^-1. 

Hence,  it  may  be  inferred  that : 

Principle  7.  —  The  total  number  of  combinations  of  n  differ- 
ent things  is  2*  —  1. 

The  above  principle  may  be  established  as  follows : 
By  §  662,  when  n  is  a  positive  integer, 

ifx  =  i,      ^.  ^  1  +  »  +  "(fril  +  ■■■  +  "C  - ')(»  -  2)-:zI 

Prin.  .3,  =\ +C'[+  C^  +  -••  +  0^  =  I  +  67„,^,. 

BXBRCISBS 

612.    1.    How  many  different  sums  may  be  paid  with  a  cent> 
a  r>-cent  piece,  a  dime,  a  quarter,  a  half-<lollar,  and  a  dollar  ? 
Solution.  Cj,ui  =  2«  -  1  =  <m. 

2.  A  man  has  10  friends.  In  how  many  ways  may  he  in- 
vite one  or  more  of  them  to  dinner  ? 

3.  How  many  different  quantities  may  be  weighed  by  weights 
of  1  oz.,  1  lb.,  i  lb.,  5  lb.,  and  10  lb.  ? 


452  PERMUTATIONS   AND   COMBINATIONS 

613.   To  find  for  what  value  of  r  the  number  of  combinations  of 
n  things  taken  r  at  a  time  is  greatest. 
Since  formula  (IV),  namely, 

^r-  1.2.3..r  '  ^^^ 

has  r  factors  in  the  numerator  and  r  factors  in  the  denominator, 
it  may  be  written, 

The  numerators  in  these  factors  begin  with  n  and  decrease 
by  1  while  the  denominators  begin  with  1  and  increase  by  1. 
The  factors,  then,  are  at  first  improper  fractions  and  at 
some  point  they  begin  to  be  proper  fractions.  Hence,  C^  is 
greatest  when  it  is  the  product  of  all  the  fractions  that  are 
greater  than  1. 

1.  When  n  is  ayi  even  number. 

In  this  case,  the  numerator  of  the  first  fraction  in  (2)  is  even 
and  the  denominator  odd,  in  the  second  the  numerator  is  odd 
and  the  denominator  even,  and  so  on  alternately ;  hence,  the 
fraction  greater  than  1,  but  nearest  to  1,  is  the  fraction  whose 
numerator  is  1  greater  than  its  denominator ;  that  is,  the  value 
of  r  must  be  such  that 

n  —  r  -h  1  =  r  -\- 1,  or  /'  =  -  • 

2.  Wlien  n  is  an  odd  number. 

In  this  case,  the  numerator  and  denominator  of  the  first 
fraction  in  (2)  are  both  odd,  of  the  second  both  even,  and  so 
on  alternately ;  hence,  the  fraction  greater  than  1,  but  nearest 
to  1,  is  the  fraction  whose  numerator  is  2  greater  than  its  de- 
nominator ;  that  is,  the  value  of  r  must  be  such  that 

,   i  .  o  n  —  1 

^,_/._|_l  =r-\-2,  or  r  =  — ^• 

Note.  —  When  n  is  odd,  since  (IV)  O"  =  Ojl  _  ,.,  there  are  two  values 
of  r  for  which  O"  is  gi-eatest,  the  other  value  being  r  =     T"     • 


PERMUTATIONS  AND  COMBINATIONS  453 

BXBRCISB8 

614.  1.  For  what  value  of  r  is  C'V"  greatest?  Find  C!?  for 
t  hat  value  of  r. 

2.  What  is  the  greatest  value  of  C*  ? 
Solve  the  following  miscellaneous  exercises : 

3.  By  permuting  the  letters  of  the  word  comUer,  how  many 
Permutations  may  be  formed 

(a)  ending  in  er  / 

(6)  with  n  as  the  middle  letter  ? 

(c)  without  changing  the  position  of  any  vowel  ? 

(d)  beginning  with  a  consonant  ? 

(e)  keeping  the  vowels  in  their  present  order  ? 

Sdooestion.  —  Since  in  (e)  the  vowels  are  to  be  kept  in  the  order 
'<,  e,  the  first  consonant  may  be  placed  successively  l)efore  o,  between 
ind  u,  between  u  and  e,  and  after  e  ;  that  is,  it  may  be  placed  in  4  ways. 

Then  the  second  consonant  may  be  placed  in  5  ways,  etc. 
Or,  since  the  vowels  are  not  interchangeable,  they  may  be  considered 

alike,  and  principle  6  may  be  applied. 

4.  A  man  has  five  coats,  six  vests,  and  eight  pairs  of  trousers. 
In  how  many  different  suits  may  he  appear? 

5.  How  many  signals  may  be  made  with  7  flags  of  different 
colors  displayed  either  singly,  or  any  number  at  a  time  arranged 
vertically  with  equal  spaces  between  them  ? 

6.  How  many  permutations  of  6  letters  may  be  formed  with 
'■'>  consonants  and  3  vowels,  if  the  vowels  are  always  given  the 
even  places? 

7.  How  many  numbers  may  be  formed  with  the  digits 
1,  2,  3,  4,  3,  2,  1,  80  that  the  odd  digits  always  occupy  the 
odd  places? 

8.  If  the  number  of  permutations  of  n  different  things  taken  6 
ii  a  time  is  equal  to  24  times  the  number  of  permutations  of 
the  same  number  of  things  taken  2  at  a  time,  find  n. 


COMPLEX  NUMBERS 


616.  The  student  has  learned  that  the  indicated  even  root 
of  a  negative  number  is  called  an  imaginary  number,  and  that 
operations  involving  such  numbers  are  subject  to  the  condition 

*^^*  ( V- 1)^  or  f,  equals  -  1,  not  +  1. 

616.  Including  all  intermediate  fractional  and  surd  values, 
the  scale  of  real  numbers  may  be  written 

..._3..._2...-1...0...  +  l--h2...4-3...,  (1) 

and  the  scale  of  imaginary  numbers,  composed  of  real  multiples 
of  +  i  and  —  i,  may  be  written 

3i 2i ^...0...+^^..-f-2^...+3^....       (2) 

Since  the  square  of  every  real  number  except  0  is  positive 
and  the  square  of  every  imaginary  number  except  0^,  or  0,  is 
negative,  the  scales  (1)  and  (2)  have  no  number  in  common 
except  0.     Hence, 

A71  imaginary  number  cannot  he  equal  to  a  real  number  nor 
cancel  any  part  of  a  real  number. 

617.  The  algebraic  sum  of  a  real  number  and  an  imaginary 
number  is  called  a  complex  number. 

2  +  3V^  1,  or  2  +  .3  i,  and  a-\-h  V—  1,  or  a  +  6i,  are  complex  numbers. 
a-  +  2  a&  V—  1  —  h'^  is  a  complex  number,  since  a^  +  2  a6  V—  1  —  6^  = 

618.  Two  complex  numbers  that  differ  only  in  the  signs  of 
their  imaginary  terms  are  called  conjugate  complex  numbers. 

a  4-  hy/  —  1  and  a  —  by/—  1,  or  a  +  hi  and  a  —  hi,  are  conjugate  com- 
plex numbers. 

454 


COMPLEX  NUMBERS  456 

619.   Operations  involving  complex  numbers. 

EXERCISES 

1.   Add3-2V^=n:and2+5V^^. 
Solution 

Since,  §  61«),  the  imaginary  terms  cannot  unite  with  the  real  terms,  the 
simplest  form  of  the  sum  is  obtained  by  uniting  the  real  and  the  imaginary 
terms  separately  and  indicating  the  algebraic  sum  of  the  results. 

3_2V^  +  2  +  5\/^T  =  (3  +  2)  +  (-2V^nr+6>/-  1) 
=  6  +  3V^^. 

Simplify  the  following : 

2.    (5  +  V^  +  (V^^-3). 


3.  (2-V-16)+(3  +  V-4). 

4.  (3-V^r8)+(4-hV^^^l8). 


5.  (V-20-Vl6)-|-(V-4o-|-V4). 

6.  (4  4-V^=^)-(2  +  V^=:4). 

7.  (3-2V^-(2-3>/^=r5). 

8.  (2-2V^4-3)-(Vi6~V-^T6). 

9.  V  -  49  -  2  -  3  V  -  4  _  V  -  1  +  6. 

10.  Expand  (a  +  ^V-^l)(a-|-6V^). 

SOLl'TION 

§  106,  (a  +  hV-T)(a  +  6>/^)=  a*  +  2ahy/-i +  iby/-  l)-» 

§  <n 5,  =a^  -{-2ahV-l  -  6*. 

11.  Kxpand  (V^  -  V^^)-. 

Solution 
(>/-,  _  v/_l)«  =  6  _  2V^=l6  +(-3) 

=  2-2V-"l5. 


456 


COMPLEX   NUMBERS 


15.  (2  +  3  0'. 

16.  (2-3i)2 

17.  (a-bif. 


Expand : 

12.  (2  +  3V^(l+V-^). 

13.  (5_V^1)(1-2V^. 

14.  (V2  +  V^(V8-V^r8). 

Show  that : 

18.  (l+V"=3)(l4-V^(l+V^  =  -8. 

19.  (-l+V^(-l  +  V-3)(-l+V^=8. 

20.  (-i  +  iV^(-i  +  iV:=^)(-i  +  iV^)  =  l. 

21.  Divide  8  +  V^  by  3  +  2 V^^. 

First  Solution 

_3V3T  +  2 
-3V3I+2 

The  real  term  of  the  dividend  may  always  be  separated,  as  above,  into 
two  parts,  one  of  which  will  exactly  contain  the  real  term  of  the  divisor. 

Second  Solution 

S+V^n:  ^   (8  +  \/3T)(3-2\/^^)    ^  26  -  13  V^^  ^  2  _  V'^1 
3  +  2\/:^      (3 +2V^=l)(3-2\/^^)  9  +  ^ 


3  +  2\ 

/-I 

2-v 

/-I 

Divide : 

22.  3  by  1  -  V^2. 

23.  2  by  1  +  V^^. 

24.  4H-V4by  2-V^^. 


25.  a^  +  b'hy  a-bV^^. 

26.  a  — bi  by  ai-^b. 

27.  (1  +  1)2  by  i_^^ 


28.    Find  by  inspection  the  square  root  of  3  +  2 V  —  10. 
SoLimoN 


3  +  2V-  10=(5-2)  +  2V5.-2  =  5  +  2V5.  -2 +(-2). 


.-.  V3  4-2>/-10=V.5  +2V5.  -2+(-2)  =  V5  +  V-2. 


COMPLEX   NUMBERS  46' 

Find  by  inspection  the  square  root  of : 


29.  4-4-2V-21.  33.    4V-3-1. 

30.  1-|-2V^^.  34.    12V^-o. 

31.  6_2V37.  36.    24V^=l^-7. 


32.    9  +  LV-22.  36.    0- -\- 2  abw  -  1  -  cr. 

37.  Verify  that  —  1  +  V— 1  and  —  1  —  V—  1  are   roots   of 
the  equation  a^-\-2x-\-2  =  0. 

38.  Expand  (i  4- iV^^)'. 

620.  Tlie  sum  and  the  product  of  two  conjugate  complex  num- 
bers are  both  real.  ^ 

For,  let  a  +  6V—  1  and  a  —  hy/^  1  be  conjugate  complex  numbers. 

Their  sum  is  2  a. 

Since  (^—1)3  =  —  1»  their  product  is, 

-^  114,  a«  -  {hy/~-  \y  =  a^-{-  h^) 

621.  If  ttco  complex  numbers  are  equal,  their  real  parts  are 
equal  and  also  their  imaginary  parts. 

For,  let  a  +  &V^  =  x  +  y V^. 

Then,  a  —  a:  =  (y  —  6)V—  1, 

which,  §  616,  is  impossible  unless  a  =  x  and  y  =  b. 

622.  If  a  -f-  6  V^^  =  0,  a  and  b  being  real,  then  a  =  0  and 

/>  =  (). 


For.  if 

a  +  6\/-l=0, 

then, 

6>/-l  =  -a, 

and,  squaring, 

-6«  =  a«; 

whence, 

aa  +  A«  =  0. 

Now,  a*  and  h-  are  both  positive ;  hence,  their  sum  cannot  be  0  un- 
less each  is  separately  0 ;  that  is,  a  =  0  and  b  =  0. 


INDEX 


(The  numbers  refer  to  pages.) 


Abscissa,  201. 
Absolute  number,  13. 
Absolute  term,  68,  285. 
Absolute  value,  25. 
Addend,  25. 

Addition,  25-27,  30-33,  61-62,  135-140, 
249-250,  269-270,  455. 

defined,  25. 

elimination  by,  178-179. 

of  complex  numbers,  455. 

of  fractions,  135-140. 

of  imaginary  numbers,  269-270,  455. 

of  monomials,  30-31. 

of  polynomials,  32-33. 

of  radicals,  249-250. 
Affected   quadratic   equations,    285-302, 
326-329,  337-349. 

defined,  285. 

general   directions   for  solving,    291. 

graphic  solution  of,  326-329,  337-338. 

solved  by  completing  square,  286-290. 

solved  by  factoring,  285. 

solved  by  formula,  290. 
Aggregation,  signs  of,  15. 
Algebra,  7. 
Algebraic   expression    of    physical    laws, 

392-393. 
Algebraic  expressions,  19-20. 
Algebraic  fraction,  126. 
Algebraic  numbers,  24,  25. 
Algebraic  representation,   16,   44-45,   69, 

84,  161-162. 
Algebraic  signs,  14-17. 
Algebraic  solutions,  8-12. 
Algebraic  sum,  25. 
Alternation,  proportion  by,  375. 
Annuities,  439-442. 
Antecedent,  369. 
Antilogarithm,  defined,  429. 

finding,  429-430. 
Arithmetical  means,  399-400. 
Arithmetical  numbers,  cube  root  of,  227- 
230. 

defined,  13. 

square  root  of,  221-224. 
Arithmetical  progressions,  394-402. 
Arithmetical  series,  defined,  394. 

last  term  of,  395-396. 

sum  of,  39&-397. 
Arrangement  of  polynomial,  58. 
Associative  law,  for  addition,  30. 

for  multiplication,  52. 
Axioms,  41,  45,  208,  215. 

Base  of  system  of  logarithms,  423,  424. 
Bino  mialformula,  211-213,  239,  417-422. 
Binomial  quadratic  surd,  256. 
square  root  of,  259-261. 


Binomial  surd,  256. 

Binomial    theorem,    211-213,    239,    41&- 

422. 
Binomials,  defined,  19. 

product  of  two,  67-68. 
Biquadratic  surd,  243. 
Braces,  15. 
Brackets,  15. 
Briggs  system  of  logarithms,  424. 

Characteristic,  424. 

Circle,  330. 

Clearing    equations    of    fractions,     155- 

161. 
Coefficients,  defined,  17. 

detached,  59. 

law  of,  for  division,  73. 

law  of,  for  multiplication,  53. 

literal,  17. 

mixed,  17. 

numerical,  17. 
Cologarithm,  432. 
Combinations,  defined,  443. 
Commensurable  numbers,  370. 
Commensurable  ratio,  370. 
Common  difference,  394. 
Common  factor,  92,  119. 
Common  multiple,  123. 
Common  system  of  logarithms,  424. 
Commutative  law,  for  addition,  30. 

for  multiplication,  52. 
Comparison,  eUmination  by,   179-180. 
Complete  quadratic,  285. 
Completing  the  square,  286-290. 

factoring  by,  345-347. 

first  method  of,  286-288. 

Hindoo  method  of,  289-290. 

other  methods  of,  288-290. 
Complex  fractions,  145-148. 
Complex  numbers,  454-457. 
Composition,  proportion  by,  376. 
Composition  and  division,  proportion  by, 

377. 
Compound  expression,  19, 
Compound  interest,  439-442. 
Condition,  equation  of,  153. 
Conjugate  complex  numbers,  454. 
Conjugate  surds,  256. 
Consequent,  369. 
Consistent  equations,  177. 
Constant,  385,  411. 
Continuation,  sign  of,  16. 
Continued  fractions,  147-148. 
Continued  proportion,  379. 
Coordinates,  201. 
Couplet,  369. 
Cross-products,  68. 
Cube,  17. 


458 


INDEX 


459 


(  ube  root.  224-230. 

defined,  IS. 

of  arithmetical  numbers,  227-230. 

of  p<jlynomials.  224-227,  240. 
(  ubic  surd,  243. 

DtMiuction,  sign  of,  16. 
Dt'fiiiitions  and  notation,  13-22. 
l)«'Kn»c,  of  expression,  119. 

of  term.  119. 
I  )riioininator,  defined,  126. 

lowest  common,  134. 
I  )f|K«iident  e<iuations,  176. 
I>»'tache<i  coefficients,  59. 
I  >i  (Terence,  common,  394. 

detine<l,  25. 

of  cubes,  factorinK,  104. 

of  even  powers,  factoring,   105-106. 

of  odd  powers,  factoring,  104-105. 

of  squares,  factoring,  97-99,  109. 

of  two  numbers.  28-29. 

.square  of,  63. 

tabular,  428. 
1  )irection  uigns,  24. 
I  >iscontinuous  curve,  332. 
Discriminant,  339. 
1  )issimilar  fractions,  defined,  134. 

re<luction    to    similar    fractions,    134- 
135. 
Dissimilar  terms,  19. 
Distributive  law,  for  division,  74. 

for  evolution,  216. 

for  involution,  208. 

for  multiplication,  55. 
Divirlend.  72. 

DiviHion.  72-87,   143-148,  238-239.  241. 
253-254.  271.  431-434.  456. 

by  logarithms,  431-434. 

<lefine<l,  72. 

of  complex  numbers,  456. 

of  fraction.s,  143-148. 

of  imaginary  numbers,  271,  456. 

of  monomials.  73-74.  238.  241. 

of  polvnomial  by  monomial,  74-76,  238. 

of  polynomial  by   polynomial,   76-80. 
239 

of  radical.s.  253-254. 

proportion  by,  376. 

special  cases  in,  81-83. 
Divisor,  72. 
Duplicate  ratio,  370. 

I  Elimination,  by  addition.  17^-17Vl. 
by  compuriHon,  179-180. 
I.v  substitution,  181. 
'  \     ut.traction,  178-179. 

I.  •   nr.i.    177. 

I. lit  ire  surd.  define<l.  244. 

re<lurtion  of  mixp<l  surd  to,  247. 
i;<luation.^.  41-44.  70.  HTi,  152-206.  240, 

262-267.  279-349,  437-438. 
aflfecte*!  qua^lratic,  285-302,  326-329. 

.337-349. 
clearing  of  fractions.  155-161. 
con.oi.stent,  177. 
defined.  8.  46.  1.53. 
dependent.  176. 
equivalent.  153. 
exponential,  437-438. 


Equations,  fractional,  152.  156-161, 
182-187,  195.  196,  264-265.  292. 
293.  294.  297.  308. 

Sraphic  solution  of,  198-206,  326-338. 
omogeneous,  312. 

identical,  152. 

impossible.  266-267. 

in  <iuadratic  form,  303-30K. 

incon.sistent.  177.  205. 

independent.  177. 

indeterminate.  176.  204. 

integral,  152. 

irrational.  262. 

linear,  153,  203. 

Uteral.  152,  160-161.  186-187,  293-294. 

members  of,  41. 

numerical,  152, 

of  condition,  153. 

of  first  clegree,  153. 

of  second  degree.  279. 

pure  tjuadratic,  279-284. 

quadratic,  279-349. 

radical,  262-267,  296-297. 

roots  of,  153. 

satisfied.  153. 

simple,  41-44,  70,  85, 152-175, 198-203. 

simultaneous,     involving     quadratics, 
309-325,  334-337.     ^ 

simultaneous    simple,     176-197,    204- 
206. 

solved  by  factoring.  115-118.  285. 

solving,  153. 

symmetrical,  310. 

system  of  simultaneous,  177. 

transposition  in,  41-44. 
Ek]uivalent  equations,  153. 
Even  root,  215. 

Evolution,  214-231.  239.  240,  241.  254- 
255,  2,59-261,  435-436.  456-457. 

by  logarithms.  435-436. 

define<l.  214. 

of  arithmetical  numbers,  221-224,  227- 
230.  231. 

of  complex  numbers,  456-467. 

of  moiiomial.s.  21&-217. 

of     pijiynomials.     218-221,     224-227. 
231.239.  240,241. 

of  radicals,  254-255,  259-261. 
Exponential  equations,  437-438. 
Exponents,  denned,  18. 

fractional.  235-241. 

law  of.  for  division,  73,  82.  232. 

law  of.  for  evolution.  216.  232. 

law  of.  for  involution.  208.  232. 

law  of.  for  multiplication.  53.  232. 

negative.  233-241. 

theory  of,  232-241. 

■ero.  233.  234,  237-241. 
Expres.<dons,  algebraic.  19-20. 

oompound,  19. 

degree  of.  119. 

fractional.  19. 

homogeneous,  60. 

integral.  20.  92. 

irrational,  242. 

raixe<l.  126. 

radical.  242. 

rational.  92,  242. 

.•ample.  19. 

symmetrical.  60. 


460 


INDEX 


Extremes,  of  proportion,  373, 
of  series,  394. 

Factor,  common,  92,  119. 

defined,  17. 

rational,  244. 

rationalizing,  256. 
Factor  theorem,  106. 
Factorial  n,  444. 
Factoring,  92-118,  240,  345-347. 

by  completing  square,  345-347. 

by  factor  theorem,  106-108. 

defined,  92. 

difference  of  cubes,  104. 

difference  of  even  powers,  105-106. 

difference  of  odd  powers,  104-105. 

difference  of  squares,  97-99,  109-110. 

equations  solved  by,  115-118,  285. 

monomials,  92. 

polynomial  squares,  109. 

polynomials,  general  directions  for,  112. 

polynomials  grouped  to  show  common 


polynomial  factor,  93-94. 
)ly         ■  '         ' 


polynomials,  whose  terms  have  com- 
mon factor,  93. 

review  of,  111-114. 

roots  by,  230. 

special  devices  for,  109-110. 

sum  of  two  cubes,  104. 

sum  of  two  odd  powers,  104-105. 

summary  of  cases.  111. 

trinomials  like  ax'  +  bx  +  c,  101-103. 

trinomials  like  x'  +  px  +  q,  100—101. 

trinomial  squares,  95-97. 
Finite  geometrical  series,  sum  of,  404-405 
Finite  number,  412, 
Finite  series,  404, 
First  degree,  equation  of,  153. 
First  member,  of  equation,  41. 

of  inequality,  361, 
First  method  of  completing  square,  286- 

288. 
Formation  of  quadratic  equations,  342- 

344. 
Formula,  binomial,  211-213,  416-422, 
Formula?,  13,  22,  171-175,  187,  283-284, 
290,  302,  358,  375,  384,  392-393,  395, 
396,    397,    400,    402,   404,   406,   407 
416,   417,   418,    419,   436,   439,   441, 
442,  444,  445,  446,  447,  450,  451. 
Fourth  proportional,  375, 
Fractional  equations,  152,  155-161,  182- 
187,  195,  196,  264-265,  292,  293,  294, 
297,  308, 
Fractional  exponents,  235-241. 
Fractional  expression,  19. 
Fractional  number,  13. 
Fractions,  126-151. 

addition  of,  135-140. 

complex,  145-148. 

continued,  147-148. 

defined,  15,  126,  268. 

dissimilar,  134. 

division  of,  143-148. 

indeterminate  in  form,  415. 

multiplication  of,  141-143, 

reduction  of,  129-135. 

signs  in,  127-129, 

similar,  134. 

subtraction  of,  135-140. 


Fulcrum,  173, 
Function  of  x,  347. 

Fundamental    property    of    imaginaries, 
268. 

General  number,  14. 

General  term  of  binomial  formula,  417, 

420-421. 
Geometrical  means,  407. 
Geometrical  progressions,  402-410. 
Geometrical  seiies,  402. 

last  term  of,  402-404. 

sura  of,  404-406. 
Graphic  solutions,  198-206,  326-338. 

of  quadratics  in  x,  326-329,  337-338, 

of  quadratics  in  x  and  y,  330-337. 

of  simple  equations,  198-206. 

of  simultaneous  simple  equations,  204- 
206. 
Graphical    representation    of    quadratic 

surd,  243. 
Graphs,  198-206,  326-338. 
Grouping,  law  of,  for  addition,  30. 

law  of,  for  multiplication,  52. 

Highest  common  factor,  119-122. 
Hindoo   method    of    completing   square, 

289-21,0. 
Homogeneous  equation,  312, 
Homogeneous  expression,  60. 
Homogeneous  in  unknown  terms,  312. 
Hyperbola,  332,  333. 

Identical  equation,  152. 

Identity,  152. 

Imaginary  numbers,  215,  268-271,  339, 

454-457. 
Impossible  equation,  266-267, 
Incommensurable  numbers,  370. 
Incommensurable  ratio,  370, 
Incomplete  quadratic,  279. 
Inconsistent  equations,  177,  205. 
Independent  equations,  177. 
Indeterminate  equation,  176,  204. 
Index,  of  power,  18. 

of  root,  18. 
Index  law,  for  division,  73. 

for  multiplication,  53. 
Induction,  mathematical,  419. 
Inequalities,  361-368. 
Inequality,  defined,  361. 
Infinite  geometrical  series,  sum  of,  405- 

406. 
Infinite  number,  347,  411,    . 
Infinite  series,  404. 
Infinitesimal,  412. 
Infinity,  347. 
Inspection,  roots  by,  230. 
Integer,  13. 

Integral  equation,  152. 
Integral  expression,  20,  92. 

reduction  of  fraction  to,  133-134. 
Interpretation,  of  results,  411-415. 

of  formsaxo.2,  «    ±,  ^     »,  412-414. 
a    0    00    0    <» 
Introducing    roots,    154,    266-267,    291, 

295. 
Inverse  ratio,  370. 
Inversion,  proportion  by,  376. 


INDEX 


461 


Involution.   207-213.   264-265.   434-435. 

bv   binomial   theorem,    211-213. 

bv  logarithms,  434-435. 

.lefine<l,    207. 

of  imaKinarie-s,  269. 

of  monomialfl,  20S-210. 

of  polynomial.s,  210-213. 

<»f  radicals,  254-2.'>.'>. 
Irrational  e<iuation,  262. 
Irrational  expression,  242. 
Irrational  number.  242.  339. 

Known  number,  14. 

:ist  term,  of  arithmetical  series,  396-396. 

of  geometrical   series.   402-404. 
Law,  associative,  for  adtlition,  30. 

associative,  f«)r  tniiltiplication.  52. 

c'otnmutative,  for  addition.  30. 

commutative,  for  multiplication,  52. 

distributive,  for  divi.sion,  74. 

di.stributive,  for  evolution,  216. 

di.stributive,  for  involution,  208. 

distributive,  for  multiplication,  55. 

index,  for  divi-^ion,  73. 

index,  for  multiplication,  53. 

of  coefficients,  for  divi.sion,  73. 

of  coefficients,  for  multiplication.  63. 

of  exponents,  for  divi.«»ion,  73,  82,  232. 

of  exjMinenta.  for  evolution,  216,    232. 

of  exponents,  for  involution,  208.  232. 

of   exponents,    for   multiplication,    53, 
232. 
I>aw  of  Kroupine,  for  addition,  30. 

of  grouping,  tor  multiplication   52. 

of  order,  for  a<ldition.  30. 

of  order,  for  multiplication,  52. 

of  signs,  for  division,  72,  S2. 

of  signs,  for  involution,  208. 

of  signs,  for  multiplication,  52. 

of  signs,  for  real  rtnits,  215. 
Lever,  173-174,  358.  390. 
Limit  of  variable,  411. 
Linear  equation,  153,  203. 
Literal  coefficient,  17. 
Literal  e<iuationB,  152.  160-161.  186-187, 
2<).3-294. 

iicfiiieil.  152. 
Literal  number.  8. 
I  .oRarithms.  423-442. 

Brigga  system  of,  424. 

characteristic  of,  424. 

'■'— •  -'tem  of.  424. 

431-434. 

i> '^^>-436. 

tindir. 

in  inT>  tns,  439-442. 

mvolu;, ,  ^.i4-435. 

mantissa  <)f,  424. 

multiplication  by,  430-431.  432-434. 

table  of,  426-427. 
I    'wer  terms,   reduction  of  fractions    to. 

130-132. 
i^owest  common  denominator,  134. 
Lowest  common  multiple,  123-126. 
Lowest  terms,  129. 

Mantissa.  424. 

Mathematical  induction,  417-419. 


Mean  proportional.  374. 
Means,  arithmetical,  399-400. 

geometrical.  407. 

of  proportion,  373. 

of  series,  394. 
Member,  of  equation,  41. 

of  inequality,  361. 
Minimum  points,  328. 
Minuend.  25. 
Mixed  coefficient,  17. 
.Mixed  expression,  126. 
Mixed  number,  126. 
Mixe<l  surd,  244. 

re<luction  to  entire  surd.  247. 
Monomials,  addition  of,  30-31. 

defined,  19. 

divi.sion  of,  73-74,  238,  241. 
Monomials,   evolution   of.   216-217.   239. 
241. 

factoring,  92. 

involution  of,  208-210,  239. 

multiplication  of,  53-55,  237,  241. 
Multiple  proportion.  378. 
Multiplicand,  51. 

Multiplication,  51-71,  141-143,  237-238. 
241.  250-252.  270-271,  4.30-431. 
432-434,  455-456. 

by  logarithms.  430-431,  432-4.34. 

defined,  51. 

of  complex  numbers,  4.55-456. 

of  fractions,  141-143. 

of  imaginarie.>4,  270-271,  455-4.56. 

of  monomials,  .53-55,  237,  241. 

of  polynomial  by  monomial,  55-56. 

of  polynomial  by  polynomial.  56-61,  62. 

of  radicals.  2,50-2,52. 

special  ca.ses  in.  63-68. 
Multiplier.  51. 

Nature  of  roots  of  quadratic  equation, 

339-.341. 
Negative  exponent,  2.33-241. 
Negative  numl)er8,  23-29,  268. 
Negative  term,  27. 
Negative  unit.  24. 
NoUtion.  13-22. 
Numl)er,  abs<ilute,  13. 

algebraic,  25. 

arithmetical,  13. 

cologarithm  of.  432. 

finite.  412. 

fractional.  13. 

general,  14. 

mfinite,  347.  411. 

infinitesimal.  412. 

irrational.  242. 

known.  14. 

literal.  8. 

logarithm  of,  423. 

mixed,  126. 

negative,  24. 

of  roots  of  (|uadratic  ecjuation,  344. 

positive,  24. 

prime.  92. 

rational,  242.  3.39. 

root  of.  18. 

unknown.  14. 

whole,  13. 
Numbers,  commensurable.  370. 

complex,  454-457. 


462 


INDEX 


Numbers,  imaginary,  215,  268-271,  454- 
457. 

incommensurable,  370. 

positive  and  negative,  23-29. 

real,  215,  268. 
Numerals,  13. 
Numerator,  126. 
Numerical  coefficient,  17. 
Numerical  equation,  152. 
Numerical  substitution,  20-22,  62,   192. 

Odd  root,  215. 

Order,  law  of,  for  addition,  30. 

law  of,  for  multiplication,  52. 

of  operations,  15. 

of  radical,  243. 

of  surd,  243. 
Ordinate,  201. 
Origin,  201. 

Parabola,  327,  331. 
Parentheses,  15. 

grouping  by,  39-40. 
removal  of,  37-38. 
Pendulum,  384,  391,  393. 
Permutations,  defined,  443. 
Permutations    and    combinations,     443- 

453. 
Physical  law,  22. 

algebraic  expression  of,  392-393. 
Plotting  points  and  constructing  graphs, 

202-203. 
Polynomial  squares,  factoring,  109. 
Polynomials,  addition  of,  32-33. 
cube  root  of,  224-227. 
defined,  19. 
division  of,  74-80. 
evolution   of,    218-221,    224-227,   231, 

239-241. 
general  directions  for  factoring,  112. 
involution  of,  210-213. 
multiplication  of,  55-61. 
square  of,  64-65. 
square  root  of,  218-221. 
Positive  and  negative  numbers,   23-29. 
Positive  number,  24. 
Positive  term,  27. 
Positive  unit,  24. 

Powers,    17-18,   207-213,    224-225,    269, 
377,  434-435. 
by  binomial  formula,  416-422. 
by  logarithms,  434-435. 
defined,  17. 
index  of,  18. 
of  >/^,  269. 
Present  value  of  annuity,  442. 
Prime  number,  92. 
Prime  to  each  other,  119. 
Principal  root,  215. 

Problems,  8-12,  22,  46-48,  71,  86-87, 
163-175,  188-192,  197,  282-284, 
298-302,  323-325,  356-360,  368, 
383-384,  389-391,  397,  400-402, 
403-404,  408-410,  436,  439-440, 
441,  442. 
area,  71,  87,  164,  171,  172,  190,  282, 
283,  298,  302,  323,  324,  357,  389, 
390. 
clock,   168. 


Problems,  commercial,  9,  10,  11,  12,  47, 
48,  71,  87,  163,  164,  167,  172.  188, 

189,  190,  191,  192,  197,  283,  298. 
299,  300,  301,  .323,  324,  325,  356,357, 
373,  383,  402,  403,  404,  410,  439-442. 

defined,  9. 

digit,  166,  190,  357. 

general  directions  for  solving,  46. 

geometrical,  10,  171-172,  284.  302,  384, 

389,  390,  391,  436. 
in  aff"ected  quadratics,  298-302. 
in  annuities,  440-442. 
in  arithmetical  progressions,  397,  400- 

402. 
in  compound  interest,  439-442. 
in  geometrical  progressions,   403-404, 

408-410. 
in  inequalities,  368. 
in     permutations    and     combinations, 

445-446,    447-449,    450,    451,    453. 
in  physics,  22,  170-175,  283-284,  302, 

325,  358,  384,  389-391. 
in  proportion,  383-384. 
in  pure  quadratics,  282-284. 
in  quadratics,  282-284,  298-302,  323- 

325 
in  review,  356-360. 
in  simple  equations,  8-12,  22,  46-48, 

71,  86-87,   163-175. 
in  simultaneous   quadratics,   323-325. 
in  simultaneous  simple  equations,  188- 

192    197. 
interest,  167,  172,  192,  325,  410,  439- 

442. 
in  variation,  389-391. 
miscellaneous,  9-12,  46,  47,  48,  71,  86, 

87,  163,  164,  166,  172,  173,  188,  189, 

191,  192,  197,  282,  298,  299,  300,  301, 

323,   324,   325,    356,   357,   358,   368, 

383,  389,   390,   391,    397,   401,    403, 

404,  409,  453. 
mixture,  169-171,  325,  356. 
percentage,    167,    170,    171,    172,    192, 

301,  325,  359,  403,  404,  410,  439-442. 
pressure,  175. 

rate,  22,  48,  71,  86,  165,  168-169,  172, 

190,  191,  192,  197,  283,  284,  300-301, 

302,  324,  325,  356,  357,  358,  389,  397, 
403-404,  408,  409. 

volume,  284,  298,  324,  389,  390,  391. 

work,  165,  191,  192,  197,  324,  357,  389. 
Product,  defined,  51. 

of  sum  and  difference  of  two  numbers, 
65-66. 

of  two  binomials,  67-68. 
Progressions,  394-410. 
Properties,  of  complex  numbers,  457. 

of  inequalities,  362-365. 

of  proportions,  374-379. 

of  quadratic  equations,  339-349. 

of  quadratic  surds,  260-261. 

of  ratios,  370-373. 
Proportion,  373-384. 

by  alternation,  375. 

by  composition,  376. 

by  composition  and  division,  377. 

by  division,  376. 

by  inversion,  376. 

continued,  379. 

defined,  373. 


INDEX 


463 


l*r<)p<»rti<>ii,  ixtjfiiio  of,  A7:\. 

means  of,  373. 

multiple,  37N. 

pmperties  of.  374-379. 
l'r<»portional,  374,  375. 
I 'lire  quadratics,  279-2H4. 

C^ua<lratic  equations,  279-349. 

affecteil.  285-302.  326-329.  337-349. 

define«l,  279. 

formation  of,  342-344. 

general  direct  ions  for  solving.  291. 

graphic  solutions  of,   326-33K. 

nature  of  r(M)ts.  339-341. 

number  of  roots,  344. 

properties  of.  339-349. 

pure.  279-2S4. 

relation  of    roots  and  coefficients,  342. 

simulUneous,  309-325.  330-337. 

solve<i  by  completinK  s<iuare.  286-290. 

8«lve<i    by    factoring.    115-118.    285. 

solved  by  formula,  290. 
'  Miatlratic    expression,    values    of,    347- 
349. 
Miadratic  form,  303. 

.Mjuations  in,  30.3-308, 
'iia<lratic  surd.  243. 

>;rapliiral  representation  of,  243. 
Quality,  signs  of.  24. 
Quotient,  72. 

Hadiral    equations,    262-267,    295-297. 

dofine<l.  262. 

general  directions  for  solving,  263. 
Radical  expression.  242. 
Radical  sign.  18. 
Ratlicals.  242-267. 

addition  of.  249-250. 

defme<l.  242. 

division  of,  253-254. 

evolution  of,  254-255. 

in  simplest  form.  244. 

involution  of,  2,54-255. 

multiplication  of,  250-252. 

order  of,  243. 

reduction  to  same  order.  248. 

reduction   to  simplest  form,   244-247. 

similar,  249. 

subtraction  of.  249-250. 
Ha.liraii.l.  242. 
l{:iti.>,  :ui'.»  :i73. 

riiriiirii'risurable.  370. 

defirMHl,  .369. 

duplirate.  370. 

incommensurable,  370. 

inverse.  370. 

of  e<|uality,  370. 

nt  geometrical  series.  402. 

of  greater  ine<juality.  370. 

oi  less  inequality.  370. 

reciprocal.  370. 

sign  of.  369. 

triplicate,  370. 
i{atio  and  proportion,  309-384. 
Katiorial  expression,  92.  242. 
Rational  factor.  244. 
Kiitioiial  numlier.  242. 
Rationalization.  2.5.5-2.58. 
Ratios,  properties  of,  370-373. 
Real  numl^ers,  215,  268. 


Reciprocal.  143. 

Reciprocal  ratio,  370. 

Rectangular  coordinates,  201. 

Reduction,  129. 

Reduction  of  fractions,   129-1.35. 

to  higher  or  lower  terms,  1.30-132. 

to  integral  or  mixed  expressions.  1.33- 
134. 

to  similar  fractions,  134-135. 
Re<luction  of  mixed  surd  to  entire  surd, 

247. 
Re<luction  of  radicals,  244-248. 

to  same  order,  248. 

to  simplest  form,  244-247. 
Relation  of  rwjts  to  coefficients  in  quad- 
ratic e<iuations.  342. 
Remainder,  25. 

Removing  roots,  154.  291,  295. 
Representation,  algebraic.   16,  44-45.  69, 

84,  161-162. 
Results,  interpretation  of,  411-415. 
Review,  49-50,  88-91,  111-114,  149-151, 

272-278,  350-360. 
Root,  cube,  18,  224-231. 

even,  215. 

index  of,  18. 

odd,  215, 

of  equation.  153. 

of  number,  18. 

principal,  215. 

square,  18,  218-224,  239,  241. 
Root  .sign,  18. 
Roots.  18,  214-231,  239-269.  4.3.5-436. 

by  factoring,  230. 

by  inspection  and  trial.  230. 

by  logarithms.  435-436. 

introduced,  154,  266-267,  '291.  295. 

nature  of.  339-341. 

number  of,  344. 

of  quadratic  equation,  280,  339-344. 

relation  to  coefficients,  342. 

removed.  154,  291,  295. 

successive  extraction  of,  231. 

Satisfying  an  e<iuation,  153. 
Scale  of  algebraic  numbers,  24. 
Second  meml>er,  of  equation.  41. 

of  inequality,  361. 
Series,  arithmetical,  394. 

defined,  394. 

extremes  of,  394. 

finite,  404. 

geometrical,  402. 

Infinite.  404. 

means  of.  394. 

terms  of,  394. 
Sign,  of  addition,  14. 

of  continuation,  16. 

of  deduction,  16. 

of  division.  14. 

of  equality.  15. 

of  inequabty.  361. 

of  infinity,  347.  411. 

of  infinitesimal.  412. 

of  multiplication,  14. 

of  product,  52, 

of  ratio.  369. 

of  subtraction,  14. 

of  variation,  385. 

radical.  18. 


464 


INDEX 


Sign,  root,  18. 
Signs,  algebraic,  14. 

direction,  24. 

in  fractions,  127-129. 

law  of,  for  division,  72,  82. 

law  of,  for  involution,  208. 

law  of,  for  multiplication,  52. 

law  of,  for  real  roots,  215. 

of  aggregation,  15. 

of  quality,  24. 
Similar  fractions,  defined,  134. 

reduction  to,  134-135. 
Similar  radicals,  249. 
Similar  terms,  19. 

Simple  equations,  41-44,  70,  85,  152-175, 
198-203. 

defined,  153. 

graphic  solution  of,  198-203. 
Simple  expression,  19. 
Simplest  form,  of  radical,  244. 

reduction  of  radicals  to,  244-247. 
Simultaneous  equations,  defined,   177. 
Simultaneous  equations  involving  quad- 
ratics, 309-325,  334-337. 

both   quadratic   and    homogeneous   in 
vmknown  terms,  313-315. 

both     quadratic,     one     homogeneous, 
312—313 

both  symmetrical,  310-312. 

division  of  one  by  other,  318. 

elimination  of  similar  terms,  319. 

graphic  solution  of,  334-337. 

one  simple,  other  higher,  309-310. 

special  devices,  315-320. 

symmetrical   except  as  to   sign,   316- 
317. 
Simultaneous  simple  equations,  176-197, 
204-206. 

graphic  solution  of,  204-206. 
Solution  of  problems,  45-48. 

defined,  9. 

general  directions  for,  46. 
Solution  of  exponential  equations,  437- 

438. 
Solutions,  graphic,  198-206,  326-338. 
Solving  the  equation,  153. 
Square,  17. 

of  any  polynomial,  64-65. 

of  difference  of  numbers,  63-64. 

of  sum  of  two  numbers,  63. 
Square  root,  218-224  239,  240,  255,  456- 
457. 

defined,  18. 

of  arithmetical  numbers,  221-224. 

of  binomial  quadratic  surds,  259-261. 

of  complex  numbers,  456-457. 

of  polynomials,  218-221,  239. 
Substitution,  defined,  20. 

elimination  by,  181. 

numerical,  20-22,  62,  192. 
Subtraction,    25,    28-29,    34-48,    61-62, 
135-140,  249-250,  269-270,  455. 

defined,  25. 

elimination  by,  178-179. 

of  complex  numbers,  456. 

of  fractions,  135-140. 


Subtraction,  of  imaginaries,  269-270,  4.55. 

of  radicals,  249-250. 
Subtrahend,  25. 

Successive  extraction  of  roots,  231. 
Sum,  of  arithmetical  series,  396-397. 

of  finite  geometrical  series    404-405. 

of  infinite  geometrical  series,  405-406. 

of  two  odd  powers,  factoring,  104-105. 

of  two  cubes,  factoring,  104. 

of  two  numbers,  square  of,  63. 

of  two  or  more  numbers,  25. 
Sum    and    difference    of   two    numbers, 

product  of,  65-66. 
Surd,  242,  268. 

binomial,  256. 

binonaial  quadratic,  256. 

biquadratic,  243. 

cubic,  243. 

entire,  244. 

mixed,  244. 

order  of,  243. 

quadratic,  243. 
Surds,  conjugate,  256. 
Symmetrical  equation,  310. 
Symmetrical  expression,  60. 
System  of  equations,  177. 

Table  of  logarithms,  426-427. 
Tablular  difference,  428. 
Term,  absolute,  68,  285. 

defined,  19,  394. 

degree  of,  119. 

negative,  27, 

positive,  27. 
Terms,  dissimilar,  19. 

lowest,  129. 

of  fraction,  126. 

of  series,  394. 

similar,  19. 
Theory  of  exponents,  232-241. 
Third  proportional,  375. 
Transposition  in  equations,  41-44. 
Trinomial,  19. 
Triplicate  ratio,  370. 

Unit,  24. 

Unknown  number,  14. 

Value,  absolute,  25. 

of  fraction  indeterminate  in  form,  415. 
Values  of  a   quadratic  expression,   347- 

349. 
Variable,  385,  411. 
Variation,  385-393. 
Vary,  385,  386. 
Velocity,  22.  172,  393. 
Vertical  bar,  15. 
Vincvdum,  15. 

Whole  number,  13. 

a:-axis,  200 

j/-axis,  200. 

Zero,  24. 

Zero  exponent,  233,  234,  237-241. 


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